Research on Offshore Foundations:
Papers at the International Symposium on Frontiers in Offshore Geotechnics Perth, Australia, 2005
by
G.T. Houlsby, C.M. Martin, B.W. Byrne, R.B. Kelly E.J. Hazell, L. Nguyen-Sy, F.A. Villalobos and L-B. Ibsen
Report No. OUEL 2275/05
University of Oxford Department of Engineering Science Parks Road, Oxford, OX1 3PJ, U.K.
Tel. 01865 273162/283300
Fax. 01865 283301 Email [email protected]
http://www-civil.eng.ox.ac.uk/
Research on Offshore Foundations: Papers at the International Symposium on Frontiers in Offshore Geotechnics
Perth, Australia, 2005
G.T. Houlsby, C.M. Martin, B.W. Byrne, R.B. Kelly E.J. Hazell, L. Nguyen-Sy, F.A. Villalobos and L-B. Ibsen
This report consists of six papers that have been accepted for the International Symposium on Frontiers in Offshore Geotechnics at Perth Australia in September 2005. The abstracts of the six papers are: a) Keynote Paper : “Suction caissons for wind turbines.” Authors: Houlsby, G.T., Ibsen, L-B. and Byrne, B.W. Abstract: Suction caissons may be used in the future as the foundations for offshore wind turbines. We review recent research work on the development of design methods for suction caissons for these applications. We give some attention to installation, but concentrate on design for in-service performance. Whilst much can be learned from previous offshore experience, the wind turbine problem poses the particularly challenging combination of a relatively light structure, with large imposed horizontal forces and overturning moments. Monopod or tripod/tetrapod foundations result in very different loading regimes on the foundations, and we consider both cases. The results of laboratory studies and field trials are reported. We also outline briefly numerical and theoretical work that is relevant. Extensive references are given to sources of further information. b) “Bearing capacity of parallel strip footings on non-homogeneous clay.” Authors: Martin, C.M. and Hazell, E.J. Abstract: On soft seabed soils, subsea equipment installations are often supported by mudmat foundation systems that can be idealised as parallel strip footings, grillages, or annular (ring-shaped) footings. This paper presents some theoretical results for the bearing capacity of (a) two parallel strip footings, otherwise isolated; (b) a long series of parallel strip footings at equal spacings. The soil is idealised as an isotropic Tresca material possessing a linear increase of undrained strength with depth. The bearing capacity analyses are performed using the method of characteristics, and the trends of these (possibly exact) results are verified by a companion series of upper bound calculations based on simple mechanisms. Parameters of interest are the footing spacing, the relative rate of increase of strength with depth, and the footing roughness. An application of the results to the design of perforated mudmats is discussed. c) “Investigating six degree-of-freedom loading on shallow foundations.” Authors: Byrne, B.W. and Houlsby, G.T. Abstract: Previous laboratory studies of the response of shallow foundations have only considered planar loading. This paper describes the development of a loading device capable of applying general loading on model shallow foundations. Loading involving all six degrees of freedom {vertical (V), horizontal (H2, H3), torsion (Q) and overturning moment (M2, M3)}, can be applied experimentally to the model foundations. Aspects of the design, including the loading rig configuration, development of a six degree-of-freedom load cell, numerical control algorithms and an accurate displacement measuring system are described. Finally results from initial experiments are presented that provide evidence for the generalisation of existing work-hardening plasticity models from planar loading to the general loading condition.
d) “The tensile capacity of suction caissons in sand under rapid loading.” Authors: Houlsby, G.T., Kelly, R.B. and Byrne, B.W. Abstract: We develop here a simplified theory for predicting the capacity of a suction caisson in sand, when it is subjected to rapid tensile loading. The capacity is found to be determined principally by the rate of pullout (relative to the permeability of the sand), and by the ambient pore pressure (which determines whether or not the water cavitates beneath the caisson). The calculation procedure depends on first predicting the suction beneath the caisson lid, and then further calculating the tensile load. The method is based on similar principles to a previously published method for suction-assisted caisson installation (Houlsby and Byrne, 2005). In the analysis a number of different cases are identified, and successful comparisons with experimental data are achieved for cases in which the pore water either does or does not cavitate. e) “Theoretical modelling of a suction caisson foundation using hyperplasticity theory.” Authors: Nguyen-Sy, L. and Houlsby, G.T. Abstract: A theoretical model for the analysis of suction caison foundations, based on a thermodynamic framework (Houlsby and Puzrin, 2000) and the macro-element concept is presented. The elastic-plastic response is first described in terms of a single-yield-surface model, using a non-associated flow rule. To capture hysteresis phenomena, this model is then extended to a multiple yield surface model. The installation of the caisson using suction is also analysed as part of the theoretical model. Some preliminary numerical results are given as demonstrations of the capabilities of the model.. f) “Moment loading of caissons installed in saturated sand.” Authors: Villalobos, F.A., Byrne, B.W. and Houlsby, G.T. Abstract: A series of moment capacity tests have been carried out at model scale, to investigate the effects of different installation procedures on the response of suction caisson foundations in sand. Two caissons of different diameters and wall thicknesses, but similar skirt length to diameter ratio, have been tested in water-saturated dense sand. The caissons were installed either by pushing or by using suction. It was found that the moment resistance depends on the method of installation.
1. INTRODUCTION The purpose of this paper is to review recent research work on the design of suction caisson foundations for offshore wind turbines. Most of the relevant work has been conducted at, or in co-operation with, the universities of Oxford and Aalborg, so we report here mainly the work of our own research groups.
Suction caissons have been extensively used as anchors, principally in clays, and have also been used as foundations for a small number of offshore platforms in the North Sea. They are currently being considered as possible foundations for offshore wind turbines. As discussed by Houlsby and Byrne (2000) and by Byrne and Houlsby (2003), it is important to realise that the loading regimes on offshore turbines differ in several respects from those on structures usually encountered in the offshore oil and gas industry. Firstly the structures are likely to be founded in much shallower water: 10m to 20m is typical of the early developments, although deeper water applications are already being planned. Typically the structures are relatively light, with a mass of say 600t (vertical deadload 6MN), but in proportion to the vertical load the horizontal loads and overturning moments are large. For instance the horizontal load under extreme conditions may be about 60% of the vertical load.
An important consideration is that, unlike the oil and gas industry where large one-off structures
dominate, many relatively small and inexpensive foundations are required for a wind farm development, which might involve anything from 30 to 250 turbines.
The dominant device used for large scale wind power generation is a horizontal axis, 3-bladed
Suction Caissons for Wind Turbines
Guy T. Houlsby1, Lars Bo Ibsen2 & Byron W. Byrne1 1: Department of Engineering Science, Oxford University, U.K. 2: Department of Civil Engineering, Aalborg University, Denmark
ABSTRACT: Suction caissons may be used in the future as the foundations for offshore wind turbines. We review recent research on the development of design methods for suction caissons for these applications. We give some attention to installation, but concentrate on design for in-service performance. Whilst much can be learned from previous offshore experience, the wind turbine problem poses a particularly challenging combination of a relatively light structure, with large imposed horizontal forces and overturning moments. Monopod or tripod/tetrapod foundations result in very different loading regimes on the foundations, and we consider both cases. The results of laboratory studies and field trials are reported. We also outline briefly relevant numerical and theoretical work. Extensive references are given to sources of further information.
Figure 1: Offshore tests in Frederikshavn, Denmark. Front: Vestas V90 3.0MW turbine. Back: Nordex 2.3MW turbine.
turbine with the blades upwind of the tower, as shown in Figure 1. The details of the generator, rotational speed and blade pitch control vary between designs. Most offshore turbines installed to date generate 2MW rated power, and typically have a rotor about 80m in diameter with a hub about 80m above mean sea level. The size of turbines available is increasing rapidly, and prototypes of 5MW turbines already exist. These involve a rotor of about 128m diameter at a hub height of about 100m. The loads on a “typical” 3.5MW turbine are shown in Figure 2, which is intended to give no more than a broad indication of the magnitude of the problem.
Figure 2: Typical loads on a 3.5MW offshore wind turbine
Note that in conditions as might be encountered
in the North Sea, the horizontal load from waves (say 3MN) is significantly larger than that from the wind (say 1MN). However, because the latter acts at a much higher point (say 90m above the foundation) it provides more of the overturning moment than the wave loading, which may only act at say 10m above the foundation. Using these figures the overturning moment of 120MNm would divide as 90MNm due to wind and 30MNm due to waves.
Realistic combinations of loads need to be considered. For instance the maximum thrust on the turbine occurs when it is generating at the maximum allowable wind speed for generation (say 25m/s). At higher wind speeds the blades will be feathered and provide much less wind resistance. It is thus unlikely that the maximum storm wave loading would occur
at the same time as maximum thrust. Turbine designers must also consider important load cases such as emergency braking. It is important to recognise that the design of a turbine foundation is not usually governed by considerations of ultimate capacity, but is typically dominated by (a) considerations of stiffness of the foundation and (b) performance under fatigue loading.
An operational wind turbine is subjected to harmonic excitation from the rotor. The rotor's rotational frequency is the first excitation frequency and is commonly referred to as 1P. The second excitation frequency to consider is the blade passing frequency, often called 3P (for a three-bladed wind turbine) at three times the 1P frequency.
Figure 3 shows a representative frequency plot of a selection of measured displacements for the Vestas V90 3.0MW wind turbine in operational mode. The foundation is a suction caisson. The measured data, monitoring system and “Output-Only Modal Analysis” used to establish the frequency plot are described in Ibsen and Liingaard (2005). The first mode of the structure is estimated, and corresponds to the frequency observed from idling conditions. The peak to the left of the first natural frequency is the forced vibration from the rotor at 1P. To the right of the first natural frequency is the 3P frequency. It should be noted that the 1P and 3P frequencies in general cover frequency bands and not just two particular values, because the Vestas wind turbine is a variable speed device.
To avoid resonances in the structure at the key excitation frequencies (1P, 3P) the structural designer needs to know the stiffness of the foundation with some confidence, this means that problems of deformation and stiffness are as important as capacity. Furthermore, much of the structural design is dictated by considerations of high cycle fatigue (up to about 108 cycles), and the foundation too must be designed for these conditions.
2. CASES FOR STUDY
The two main problems that need to be studied in design of a suction caisson as a foundation are:
• installation; • in service performance.
In this review we shall discuss installation methods briefly, but shall concentrate mainly on design for in service performance. The relevant studies involve techniques as diverse as laboratory model testing, centrifuge model testing, field trials at reduced scale, and a full-scale field installation.
Complementing these experiments are numerical studies using finite element techniques, and the development of plasticity-based models to represent the foundation behaviour.
Suction caissons may be installed in a variety of soils, but we shall consider here two somewhat idealised cases: a caisson installed either in clay, which may be treated as undrained, or in sand. For typical sands the combination of permeability value, size of caisson and loading rates leads to partially drained conditions, although much of the testing we shall report is under fully drained conditions. In this paper we report mainly work on sands.
We shall consider two significantly different loading regimes, which depend on the nature of the structure supporting the wind turbine. Most offshore wind turbines to date have been supported on a “monopile” – a single large diameter pile, which in effect is a direct extension of the tubular steel tower which supports the turbine. Some turbines have been supported on circular gravity bases. An obvious alternative is to use a single suction caisson to support the turbine, and we shall call this a “monopod” foundation, Figure 4(a). The monopod resists the overturning moment (usually the most important loading component) directly by its rotational fixity in the seabed.
As turbines become larger, monopod designs may become sufficiently large to be uneconomic, and an alternative is a structure founded on three or four smaller foundations: a “tripod” or “tetrapod”, Figure
4(b). In either of these configurations the overturning moment on the structure is resisted principally by “push-pull” action of opposing vertical loads on the upwind and downwind foundations. Alternatives using asymmetric designs of tripod, and those employing “jacket” type substructures are also under consideration.
(a) (b) Figure 4: caisson foundations for a wind turbine, (a) monopod, (b) tripod/tetrapod
1-100
-80
-60
-40
-20
0
20
Frequency Domain Decomposition - Peak PickingAverage of the Normalized Singular Values of
Spectral Density Matrices of all Data Sets.dB | 1.0 / Hz
Frequency
First mode3P
1P
Figure 3. Frequency plot of measured displacements for a wind turbine in operational mode.
3. NORMALISATION PROCEDURES
A number of studies have been conducted at different scales and it is necessary to compare the results from these various studies. To do this it is appropriate to normalise all the results so that they can be represented in non-dimensional form. This procedure also allows more confident extrapolation to full scale.
The geometry of a caisson is shown in Figure 5. The outside radius is R (diameter oD ), skirt length is L and wall thickness t. In practice caissons may also involve stiffeners on the inside of the caisson, these being necessary to prevent buckling instability during suction installation, but we ignore these in a simplified analysis. Geometric similarity is achieved by requiring similar values of RL 2 and Rt 2 .
Figure 5: Geometry of a caisson foundation
Figure 6: Loading and displacement conventions for a caisson foundation (displacements exaggerated).
The sign convention for applied loads and
displacements is shown in Figure 6. The rotation of the caisson θ is already
dimensionless, and we normalise the displacements simply by dividing by the caisson diameter, to give
Rw 2 and Ru 2 .
In sand it is straightforward to show that, for similar values of dimensionless bearing capacity factor, the loads at failure would be proportional to γ′ and to 3R . We therefore normalise vertical and
horizontal loads as γ′π 32 RV and γ′π 32 RH , where we have included the factor π2 to give the normalisation factor a simple physical meaning: it is the effective weight of a cylinder of soil of the same diameter of the caisson, and depth equal to the diameter. In a similar way we normalise the overturning moment as γ′π 44 RM .
Use of the above normalisation is appropriate for comparing tests in sands with similar angles of friction and dilation. We recognise that these angles both decrease slightly with pressure and increase rapidly with Relative Density (Bolton, 1986). This means that comparable tests at smaller scales (and therefore lower stress levels) will need to be at lower Relative Densities to be comparable with field tests.
In clay the vertical capacity is proportional to a representative undrained shear strength us and to
2R , so we normalise loads as usRV 2π and
usRH 2π , and the moment as usRM 32π . In order to be comparable, tests at different scales
will need the profile of undrained strength with depth to be similar. If the strength profile is fitted by a simple straight-line fit zss uou ρ+= , then this requires similar values of the factor uosRρ2 .
Scaling of results using the above methods should give satisfactory results in terms of capacity. For clays it should also lead to satisfactory comparisons in terms of stiffness, provided that the clays being compared have similar values of ur sGI = . This condition is usually satisfied if the clays are of similar composition and overconsolidation ratio. For sands, however, an extra consideration needs to be taken into account. The shear modulus of a sand does not increase in proportion to the stress level, but instead can reasonably be expressed by:
n
aa ppg
pG
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′= (1)
where g and n are dimensionless constants, and ap is atmospheric pressure (used as a reference pressure). The value of n is typically about 0.5, so that the stiffness is proportional roughly to the square root of pressure.
Comparing rotational stiffnesses on the basis of a plot of γ′π 44 RM against θ effectively makes the assumption that the shear stiffness is proportional to
γ′R2 , which may be regarded as a representative stress level. Since in fact the stiffness increases at a lower rate with stress level, this comparison will result in larger scale tests giving lower apparent normalised stiffness. This effect can be reduced by multiplying the θ scale by the dimensionless factor ( ) n
a Rp −γ′ 12 , which compensates for the stiffness variation with stress level.
Thus we recommend that to compare both stiffness and capacity data for sands one should plot
γ′π 44 RM against ( ) 5.02 γ′θ Rpa (assuming
5.0=n ) for moment tests, and γ′π 32 RV against
( )( ) 5.022 γ′RpRw a for vertical loading tests. A fuller description of these normalisation procedures is given by Kelly et al. (2005a).
4. INSTALLATION STUDIES
The principal difference between installation of a suction caisson for an offshore wind turbine and for previous applications is that the turbines are likely to be installed in much shallower water. There is a popular misconception that suction caissons can only be installed in deep water, where a very substantial head difference can be established across the lid of the caisson. In shallow water the net suction that can be achieved is indeed much smaller (being limited by the efficiency of the pumps, as the absolute pressure approaches zero), but the suctions that can be achieved are nevertheless sufficient for installation in most circumstances. Only in stiff clays is it likely that some possible caisson designs, which might otherwise be suitable as far as in-service conditions are concerned, could not be installed by suction in shallow water.
In Table 1 we list the main instances where caissons have been installed in shallow water, as appropriate to wind turbine installations. The water depths wh are approximate only. In addition to the field tests listed, a large number of small scale model tests of installation have been carried out at Oxford University (on caissons of 0.1m to 0.4m diameter), the University of Western Australia (UWA), Aalborg and elsewhere.
The largest completed installation in shallow water is that of a prototype suction caisson, shown in Figure 7, installed in the offshore research test facility in Frederikshavn, Denmark. The prototype
has a diameter of 12m and a skirt length of 6m. The operational water depth is 4m, and as the site is in a basin, no wave or ice loads are applied. As seen in Figure 7 the suction caisson was installed in only 1m of water in the basin. The steel construction has a mass of approximately 140t, and the caisson was placed in late October 2002. The installation period
Table 1: Installations in shallow water
Site Soil wh(m)
D (m)
L (m) Ref.
Wilhelmshaven Sand 6.0 16.0 15.0 Installation April 2005
Frederikshavn Sand 1.0 12.0 6.0 30
Frederikshavn Sand 0.2 2.0 4.0
2.0 4.0
-
Sandy Haven Sand 0.5 4.0 2.5 23 Tenby Sand 2.0 2.0 2.0 23 Burry Port Sand 0.5 2.0 2.0 -
Luce Bay Sand 0.2 3.0 1.5
1.5 1.0 27
Bothkennar Clay 0.2 3.0 1.5
1.5 1.0
26
(a)
(b)
Figure 7: Installation of the prototype foundation at the test site in Frederikshavn: (a) during installation, (b) at the end of installation.
was about 12 hours, with the soil penetration time being 6 hours. A computer system was used to control the inclination, suction pressure and penetration rate. Det Norske Veritas (DNV) has certified the design of the prototype in Frederikshavn to B level. The Vestas V90 3.0MW turbine was erected on the foundation in December 2002. The development of the design procedure for the bucket foundation is described in Ibsen and Brincker (2004). An even larger installation is currently in progress at Wilhelmshaven, Denmark.
There are two main ways of predicting firstly the self-weight penetration of the caisson and secondly the suction required to achieve full installation. The first method (Houlsby and Byrne, 2005a,b) involves use of adaptations of pile capacity analysis, in which the resistance to penetration is calculated as the sum of an end bearing term on the rim and friction on the inside and outside. In sands the seepage pattern set up by the suction processes alters the effective stress regime in a way that aids installation.
The calculation has been implemented in a spreadsheet program SCIP. Figure 8 shows for example a comparison between variation of measured suction in a model test installation with tip penetration of the caisson (Sanham, 2003), and the SCIP calculation.
0
50
100
150
200
250
0 500 1000 1500 2000 2500 3000Suction, s (Pa)
Pene
tratio
n, h
(mm
)
SCIP ResultsExperimental Result
Figure 8: Comparison of SCIP with model test
The other approach involves use of CPT data to infer directly the resistance dR to penetration of the caisson. The required suction requ to penetrate the caisson to depth d is calculated as:
( ) '( )( ) dreq
suc
R d G du dA−
∆ = (2)
where '( )G d is the self-weight of the caisson at penetration depth d (reduced for buoyancy), and
sucA is the area inside the caisson, where the suction is applied.
The penetration resistance is calculated from the following expression, which is based on calibration against measured data:
0
0
( ) ( ) ( ) ( ) ( )
( ) ( )
d
d t tip t out out s
d
in in s
R d K d A q d A K z f z dz
A K z f z dz
= + +∫∫
(3)
where tq is the corrected cone resistance and sf the sleeve friction at depth z. tK is a coefficient relating
tq to the unit tip resistance on the rim. This resistance is adjusted for the reduction due to the applied suction by the expression:
1t
t t tcrit
uK k ru
β⎛ ⎞∆
= −⎜ ⎟∆⎝ ⎠ (4)
where tk is an empirical coefficient relating tq to the tip resistance during static penetration of the caisson,
tr is the maximum reduction in tip resistance. critu∆ is the critical suction resulting in the critical hydraulic gradient 1criti = along the skirt. tβ is an empirical factor.
outK and inK are coefficients relating sf to the unit skin friction on the outside and inside of the skirt. The water flow along the skirt changes the skin friction. For the inside skin friction the coefficient reduces the skin friction when suction is applied, whereas on the outside the skin friction is increased. The coefficients are established as:
1
1
out
in
out out outcrit
in in incrit
uK ru
uK ru
β
β
α
α
⎛ ⎞∆= +⎜ ⎟∆⎝ ⎠
⎛ ⎞∆= −⎜ ⎟∆⎝ ⎠
(5a,b)
Where outα and inα are empirical coefficients relating
sf to the unit skin friction during static penetration of the caisson. outr and inr are the maximum changes in skirt friction. outβ and inβ are empirical factors.
The required suction ureq to penetrate the prototype in Frederikshavn was predicted using equation (2). The result of the analysis is shown in Figure 9. The lower line represents ureq calculated from the CPT tests. The curved line represents the limiting suction upip which would cause piping to occur. umax is the theoretical maximum net suction, limited by the possibility of cavitation within the caisson, as the absolute pressure approaches zero, so
that =maxu 100kPa above water level and increases linearly with the water depth, as shown by Figure 9. umax is used to calculate the accessible net suction, which is limited by the efficiency of the pumps, upump. As is seen, the suction in shallow water can be limited either by the suction causing piping or by the accessible net suction available from the pumps.
The suction upip causing piping has been studied at the test site in Frederikshavn by installation tests on 2x2m and 4x4m caissons. Figure 10 shows a 4x4m caisson where the limiting suction upip has been achieved, and soil failure by piping has occurred. The soil outside of the skirt is sucked into the caisson and the penetration of the caisson cannot proceed.
If a tripod or tetrapod structure is to be installed, then levelling of the structure can be achieved by separately controlling the suction in each of the caissons. For a monopod structure, however, an alternative strategy has to be adopted. Experience suggests that for installation in either clay or sand, the level of the caisson is rather sensitive to the
application of eccentric loads (moments), especially in the early stages of installation. This offers one possibility for controlling the level of the caisson: by use of an eccentric load that can be adjusted in position to keep the caisson level.
An alternative strategy, which has proven to be highly successful for installation in sand, is to divide the rim into sections and to control the pressures at the skirt tip in each section individually. By applying pressure over one segment of the caisson rim the upward hydraulic gradient within the caisson can be enhanced locally, thus encouraging additional downward movement for that sector. By controlling the pressures at a number of points the caisson may be maintained level.
This method would not be applicable in clays. One possibility, as yet untried at large scale, for controlling level in clays would be to use a segmented caisson in which the suctions in the different segments could be controlled independently.
Some preliminary small scale tests suggest that this approach might be successful in sand too (Coldicott, 2005). Figure 11 shows the volumes of water pumped from the two halves of a 400mm diameter caisson split by a diametral vertical wall. About 60% of the water pumped represents the volume displaced by the descending caisson, whilst about 40% represents seepage beneath the caisson rim. Figure 12 shows that during the installation the suctions developed in the two halves were (as would be expected in a uniform material) almost equal.
5. CAISSON PERFORMANCE: MONOPOD
A large number of tests have been devoted to studying the performance of a caisson under moment loading at relatively small vertical loads, as is relevant to the wind turbine design. Some details of the test programmes are given in Table 2.
Figure 9: Suction required for installation at Frederikshavn
Figure 10: The limiting suction upip has been achieved and soil failure by piping has occurred.
0
50
100
150
200
250
300
350
400
0 5 10 15 20 25 30 35Volume, (10-3 m3)
Pen
etra
tion,
h (m
m)
Cell 1Cell 2
Total Volume
Volume Displaced
Seepage Volume
Figure 11: Volumes pumped from 2-cell caisson in sand.
5.1 Sand: field tests
The largest test involves the instrumented Vestas V90 3.0MW prototype turbine at Frederikshavn, Denmark. The caisson is installed in a shallow 4m depth lagoon next to the sea, and the turbine is fully operational. The only significant difference between this installation and an offshore one is that the structure is not subjected to wave loading.
The test program involving the prototype (turbine and caisson) is focusing on long-term deformations, soil structure interaction, stiffness and fatigue. The prototype has been equipped with: • an online monitoring system that measures the
dynamic deformation modes of the foundation and the wind turbine,
• a monitoring system that measures the long-time deflection and rotation of the caisson
• a monitoring system that measures the pore pressure along the inside of the skirt.
The online monitoring system that measures the modes of deformation of the foundation and wind turbine involves 15 accelerometers and a real-time data-acquisition system. The accelerometers are placed at three different levels in the turbine tower and at one level in compartments inside the caisson foundation. The positions are shown in Figure 13, and the locations and measuring directions are defined in Figure14.
Output-only Modal Analysis has been used to analyze the structural behaviour of the wind turbine during various operational conditions. The modal analysis has shown highly damped mode shapes of the foundation/wind turbine system, which the present aero-elastic codes for wind turbine design cannot model. Further studies are to be carried out with respect to soil-structure interaction. A detailed description of the measuring system and the Output-Only Modal Analysis is given by Ibsen and Liingaard (2005).
The static moment tests referred to in Table 2 at Sandy Haven and at Burry Port were relatively straightforward, with very simple instrumentation, but those at Frederikshavn test site and at Luce Bay were detailed investigations.
The large scale tests at Frederikshavn is part of a research and development program concerning caisson foundation for offshore wind turbines. The research program is a co-operation between Aalborg
Table 2: Moment loading tests
Site Soil D (m)
L (m) Ref.
Frederikshavn Sand 12.0 6.0 - Frederikshavn Sand 2.0 2.0 - Sandy Haven Sand 4.0 2.5 - Burry Port Sand 2.0 2.0 21 Luce Bay Sand 3.0 1.5 27
Oxford laboratory Sand
0.1 0.15 0.15 0.2 0.2 0.3
0.0-0.066 0.05 0.1 0.1 0.2
0.15
2,4 2,7
42,43 34,42,43
11,43 11,42,43
Aalborg laboratory Sand 0.2 0.3 0.4
0.0 – 0.2 0.0 – 0.3 0.0 – 0.4
-
Bothkennar Clay 3.0 1.5 26
Oxford laboratory Clay 0.2 0.3
0.1 0.15 34
UWA centrifuge (100g) Clay
0.06
0.02 0.03 0.06
12
Level IV: 89 m
Level III: 46 m
Level II: 13 m
Level I: 6 m
Figure 13: Sensor positions in tower and foundation.
0
50
100
150
200
250
300
350
400
0 1000 2000 3000 4000 5000 6000Suction, s (Pa)
Pene
tratio
n, h
(mm
)
Cell 1Cell 2
Figure 12: Suctions required for installation of 2-cell caisson in sand.
University and MBD offshore power (Ibsen et al. 2003). The large scale tests are complemented by laboratory studies. The laboratory and large scale tests are intended to model the prototype in Frederikshavn directly. In order to design a caisson foundation for offshore wind turbines several load combinations have to be investigated. Each load combination is represented by a height of load h above the foundation and a horizontal force H. The moment at the seabed is calculated as M = hH. Table 3 shows that the resulting loading height varies from 10m (for a wave force in shallow water) to 104.4m (force from normal production of a 3MW turbine in 20m of water). Scaling of the tests is achieved by:
p
mpm
DDhh = (6)
where D is the diameter of the caisson and index m and p are for model and prototype. The values of the loading height in the test program are shown in Table 3.
The large scale tests at Frederikshavn employ loading by applying a horizontal load at a fixed height, under constant vertical load. A steel caisson with an outer diameter of 2m and a skirt length of 2m has been used. The skirt is made of 12mm thick steel plate. Figure 15 shows the caisson prior to installation, and Figure 16 the overall test setup. Currently 10 experiments have been conducted, but the testing program is ongoing. Each test has three phases:
1. Installation phase: The caisson is installed by means of suction. CPT tests are performed before and after installation of the caisson.
2. Loading phase: An old tower from a wind turbine is mounted on top of the caisson. The caisson is loaded by pulling the tower
Figure 14: Sensor mountings in the tower and foundation at Frederikshavn.
Table 3. Loading heights in the Aalborg test program
Prototype Laboratory Model Field Model
m12=pD m2.0=mD 0.3m 0.4m 2.0m
ph [m] mh [m] mh [m]
104.4 1.74 2.61 3.48 17.40 69.6 1.16 1.74 2.32 11.60 38.0 0.63 0.95 1.27 6.33 20.0 0.33 0.50 0.67 3.33 10.0 0.17 0.25 0.33 1.67
Figure 15: Caisson for large scale test at Frederikshavn
loading tower
loading wiretower located on
bucket foundation
3 MW Vestaswindmill on bucket
foundation
Figure 16: Setup for combined loading of 2x2m caisson at Frederikshavn. (Back: prototype 3MW Vestas wind turbine on the 12 x 6m caisson)
horizontally with a wire. The combined loading (H,M) is controlled by changing the height of loading.
3. Dismantling phase. The caisson is removed by applying overpressure inside the bucket.
Figure 17 shows the moment rotation curve for a test on the 2x2m caisson at Frederikshavn. The test is performed with hm = 17.4m and a vertical load on the caisson of 37.3kN. The fluctuations in the curve are caused by wind on the tower.
Figure 17: Moment-rotation test on 2x2m caisson.
Tests at Luce Bay were designed by Oxford University and conducted by Fugro Ltd.. The moment loading tests were of two types. Firstly small amplitude (but relatively high frequency) loading was applied by a “Structural Eccentric Mass Vibrator” (SEMV) in which rotating masses are used to apply inertial loads at frequencies up to about 12Hz. Secondly larger amplitude, but lower
frequency, cycles were applied using a hydraulic jack. A diagram of the loading rig, which allowed both moment and vertical loading tests, is shown in Figure 18.
The SEMV test involve cycles of moment loading at increasing amplitude as the frequency increases. Figure 19 shows the hysteresis loops obtained from a series of these cycles at different amplitudes. As the cycles become larger the stiffness reduces but hysteresis increases. The tests were interpreted (Houlsby et al., 2005b) using the theory of Wolf (1994), which takes account of the dynamic effects in the soil, and the equivalent secant shear modulus for each amplitude of cycling determined.
Figure 20 shows the moment rotation curves for much larger amplitude cycling applied by the hydraulic jack. Again hysteresis increases and secant stiffness decreases as the amplitude increases. The unusual “waisted” shape of the hysteresis loops at very large amplitude is due to gapping occurring at the sides of the caisson.
The secant stiffnesses deduced from both the SEMV tests and the hydraulic jacking tests are combined in Figure 21, where they are plotted against the amplitude of cyclic rotation. It is clear that the two groups of tests give a consistent pattern of reduction of shear modulus with strain amplitude, similar to that obtained for instance from laboratory tests.
5.2 Sand: laboratory tests
Turning now to model testing, a large number of tests have been carried out both at Aalborg and at Oxford. Almost all the model tests have involved “in plane” loading (in which the moment is about an
4000 4000 6000
30001500
H
H
HB
CC
A
RR
W
V
A
B
C
V
LL
L
L L LL
(a) (b)
Figure 18: Field testing equipment, dimensions in mm. Water level and displacement reference frames not shown. (a) arrangement for jacking tests on 1.5m and 3.0m caissons, (b) alternative arrangement during SEMV tests. Labels indicate (A) A-frame, (B) concrete block, (C) caissons, (H) hydraulic jacks, (L) load cells, (R) foundations of reaction frame, (V) SEMV, (W) weight providing offset load for SEMV tests.
axis perpendicular to the horizontal load). However, a test rig capable of applying full 6 degree-of-freedom loading has recently been developed by Byrne and Houlsby (2005).
The model tests at Aalborg are performed by the test rig shown in Figure 22. The rig consists of a test box and loading frame. The test box consists of a steel frame with an inner width of 1.6m x 1.6m and an inner total depth of 0.65m. The test box is filled with Aalborg University Sand No 0. After each experiment the sand in the box is prepared in a systematic way to ensure homogeneity within the box, and between the different test boxes. The sand
is saturated by the water reservoir shown in Figure 22. Before each experiment CPT-tests are performed to verify the density and strength of the sand. The caisson is then installed and loaded with a constant vertical load. The vertical load is kept constant through the experiment, while the horizontal force is applied to the tower by the loading device mounted on the loading frame, see Figure 21. The tower and the loading device are connected by a wire. The combined loading (H, M) is controlled by the height of loading h. The loading frame allows the possibility of changing h from 0.1m to 4.0m above the sand surface (Table 3). The horizontal force H is measured by a transducer connected to the wire. The deformation of the foundation and the moment are measured with the measuring cell mounted on the top of the caisson, as shown by Figure 23.
Laboratory tests at Oxford University have used a versatile 3 degree-of-freedom loading rig designed by Martin (1994) and adapted by Byrne (2000) (see also Martin and Houlsby (2000) and Gottardi et al. (1999)). The rig is shown in Figure 24, and is capable of applying a wide range of combinations of vertical, horizontal and moment loading under either displacement or load control.
Typical moment loading tests involve applying a fixed vertical load, and then cycling the rotation at
-30
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-10
0
10
20
30
-0.00005 -0.000025 0 0.000025 0.00005
Rotation (radians)
Mom
ent (
kNm
)
6Hz
7Hz
8Hz
9Hz
10H
Figure 19: Hysteresis loops from SEMV tests on 3m caisson.
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0
100
200
300
400
500
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Rotation of caisson centre (2R θ) (m)
Mom
ent (
kNm
)
Figure 20: Hysteresis loops from hydraulic jacking tests on 3m caisson.
0
10
20
30
40
50
60
70
80
90
100
0.000001 0.00001 0.0001 0.001 0.01 0.1∆θ (radians)
G (M
Pa)
JackingSEMVHyperbolic curve fit
Figure 21: Shear modulus against rotation amplitude.
Figure 22: The caisson test rig at Aalborg University
increasing amplitude. An example is given in Figure 25.
The first interpretation of such tests is to determine the yield surface for a single surface plasticity model (see section 7.2 below, and also Martin and Houlsby (2001), Houlsby and Cassidy (2002), Houlsby (2003), Cassidy et al. (2004)). An example of the yield points obtained, plotted in the vertical load-moment plane, is given in Figure 26. Of particular importance is the fact that at very low vertical loads there is a significant moment capacity, and that this extends even into the tensile load range. In these drained tests the ultimate load in tension is a
significant fraction of the weight of the soil plug inside the caisson.
Sections of the yield surface can also be plotted in H-M space as shown in Figure 27, where the data here have been assembled from many tests at different stress levels. The flow vectors are also plotted in this figure, and show that in this plane (unlike the V-M plane) associated flow is a reasonable approximation to the behaviour. Feld (2001) has observed similar shapes of a yield surface for a caisson in sand.
We now consider the possibility of scaling the results of laboratory tests to the field. The test at Frederikshavn shown in Figure 17 was on a caisson with a ratio 12 =RL , at an RHM 2 value of
approximately 8.7, and with a value of γ′π 32 RV of about 0.62. Using the data from the Oxford laboratory on 0.2x0.2m caissons this requires a vertical load of about 60N. In fact a test had been carried out with 12 =RL and N50=V . According to the scaling relationships discussed in section 3, the moment should be scaled according to γ′4R (a factor of 6250) and the rotational displacement θR2
according to γ′3R (a factor of 25). Figures 26 and 27 suggest that for a vertical load of 60N rather than
Figure 23: The measuring cell connecting the caisson and the tower.
Figure 24: Three degree-of-freedom testing rig at Oxford University
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0
20
40
60
80
100
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0Rotational Displacement, 2Rθ (mm)
Mom
ent L
oad,
M/2
R (N
)
Figure 25: Moment-rotation test on sand
0
20
40
60
80
100
-160 -120 -80 -40 0 40 80 120Vertical Load, V (N)
Mom
ent L
oad,
M/2
R (N
)Experiment, M/2RH = 1
Fitted Yield Surface
Soil Plug Weight
Figure 26: Experimentally determined yield surface in V-M plane
50N a moment capacity say 5% higher might be expected, and that for the higher value of RHM 2 a further increase of say 15% is appropriate. We therefore apply a factor of 7500 to the moments and 25 to the rotational displacements. The result is shown in Figure 28. It can be seen that after scaling the moment at a θR2 value of 0.04 m is about 120kNm, compared to about 280kNm measured in the field. Although there is a factor of about 2 between these values, it must be borne in mind that there are a number of possible causes of difference between the tests (e.g. the sand in the field test may be much denser), and also that a factor of 7500 has already been applied: a factor of 2 is relatively small by comparison.
5.3 Clay: field and laboratory tests
Less work has been carried out on clay than on sand. The large scale trials at Bothkennar (Houlsby et al. 2005b) are complemented by laboratory studies intended to model these trials directly, and therefore add confidence to the scaling of the results to prototype size caissons (Kelly et al., 2005a).
At Bothkennar, moment loads were applied to a 3m x 1.5m caisson by two means. Small amplitude, but relatively high frequency (10Hz) loading was applied by means of the SEMV device described above, and larger amplitude cycles, but at much lower frequency, were applied using a hydraulic jack. In both cases the loading was 4m above the caisson, so that 33.1=Dhload . The most important observation from these tests was the gradual reduction of secant stiffness (and increase in hysteresis) as the amplitude of the load cycles increases.
The laboratory tests, specifically modelling the field tests, involved just relatively low frequency loading. After the scaling relationships described in section 3 were applied, there was a satisfactory agreement between laboratory and field data, especially at relatively small amplitudes of movement. As an example, Figure 29(a) shows the results (in dimensionless form) for rotation of the 3.0m diameter caisson in the field, and Figure 29(b) the equivalent results, also in dimensionless form, from the small scale model test. The pattern of behaviour is remarkably similar in the two tests. This sort of comparison is vital to establish
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0
40
80
120
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Incremental Horizontal Displacement, du (mm)
Mom
ent L
oad,
M/2
R(N
) In
crem
enta
l Rot
atio
n, 2
Rdt
heta
(mm
) V = -50 N
V = 0 N
V = 50 N
Figure 27: Yield surfaces and flow vectors in H-M space.
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0
50
100
150
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
Rotational Displacement, 2Rtheta (m)
Mom
ent,
M (k
Nm
)
Figure 28: Laboratory moment test scaled to field conditions for comparison with Figure 17
-0.4
-0.3
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-0.1
0
0.1
0.2
0.3
-0.015 -0.01 -0.005 0 0.005 0.01
θ
M/[s
u(2R
)3 ]
(a) field test
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-0.1
0
0.1
0.2
0.3
-0.015 -0.01 -0.005 0 0.005 0.01
θ
M/[s
u(2R
)3 ]
(b) model test Figure 29: Moment-rotation results presented in non-dimensional form for laboratory and field tests.
confidence in the use of model testing to develop design guidelines.
6. CAISSON PERFORMANCE: TETRAPOD OR TRIPOD
In the following, in which we consider multiple footing designs to support the wind turbine, we shall refer principally to a tetrapod (four footings) rather than a tripod. As a tripod is perhaps the most obvious multiple footing design to use, and has the obvious advantage of simplicity, our preference for the tetrapod deserves some explanation.
As is discussed below, prudent design of a multiple footing structure will avoid tension being applied to any of the foundations (except under the most extreme of circumstances). This in effect dictates the separation of the foundations for a given overturning moment and weight of structure. Approximate calculations indicate that the tetrapod structure is usually a more favourable configuration to avoid tension, as it requires somewhat less material. The differences are not large, and a tripod may be preferred in some circumstances, but we shall refer to a tetrapod, as this will probably be more efficient. The important mechanism is the same in both cases: the overturning moment is resisted by opposing “push-pull” action on the foundations.
In Table 4 we list the tests that have been carried out on vertical loading of caissons relevant to the wind turbine problem. In addition to these studies there are a number of other relevant studies which have been directed towards vertical loading of caissons for structures in the oil and gas industry or for use as anchors.
6.1 Sand: field and laboratory tests
The simplest tests on vertical loading of caissons in sand, which are relevant both to installation and to subsequent performance, simply involve pushing caissons vertically into sand to determine the vertical load-displacement response. Figure 30 shows the results of a set of such tests on caissons of different L/D ratios, Byrne et al. (2003). It is clear from the figure that there is a well-established pattern. While the caisson skirt is penetrating the sand there is relatively low vertical capacity, but as soon as the top plate makes contact with the sand there is a sudden increase in capacity. The envelope of the ultimate capacities of footings of different initial L/D ratios also forms a single consistent line.
Of most importance, however, is the performance of the caissons under cyclic vertical loading. Figure
31 shows the results of tests on a 300mm diameter caisson subjected to rapid cyclic loading. Small-amplitude cycles show a stiff response, with larger cycles showing both more hysteresis and more accumulated displacement per cycle. The most important observation is that as soon as the cycles go into tension, a much softer response is observed, and the hysteresis loops acquire a characteristic “banana” shape. Clearly the soft response on achieving tension should be avoided in design. Closer examination of the curves reveals that the softening in fact occurs
Table 4: Vertical loading tests
Site Soil D (m)
L (m)
Ref.
Luce Bay Sand 1.5 1.0 27
Oxford laboratory Sand
0.05 0.1
0.15 0.15 0.2
0.28
0.0 - 0.1 0.0 - 0.066
0.05 0.1
0.133 0.18
11 2,5 2,5 34 34
25,32,33,35 Bothkennar Clay 1.5 1.0 26 UWA centrifuge (100g)
Clay 0.06 0.02 0.03 0.06
3
0
50
100
150
200
250
300
350
400
0 50 100 150Vertical Displacement, w (mm)
Ver
tical
Loa
d, V
(N)
0 0.5 1 1.5 2 2.5 3 3.5Normalised Vertical Displacement, w/D
Figure 30: Vertical load-penetration curves for caissons of different L/D ratios
-400-200
0200400600800
1000120014001600
200 210 220 230 240 250 260 270Vertical Displacement (mm)
Verti
cal S
tress
(kP
a)
Figure 31: Cyclic vertical loading of model caisson.
once the drained frictional capacity of the skirts has been exceeded, rather than simply the transition into tension.
Paradoxically, although additional accumulated displacement is observed once tension is reached, this accumulated displacement is downwards (not upwards as one might expect because of the tensile loading).
The above observations mean that tension must be avoided in a prudent design of a tripod or tetrapod foundation for a wind turbine. However, in all but the shallowest of water, avoiding this tension means that either the foundation must have a large spacing between the footings, or that ballasting must be used. The latter may in fact be a cost effective measure in deep water.
Some designers may wish to reduce conservatism by allowing for the possibility of tension under extreme circumstances. It is therefore useful to examine the ultimate tensile capacity under rapid loading. Figure 32 shows the result of three such tests. The slowest test (at 5mm/s) is almost drained, and a very low capacity in tension is indicated. The
capacity in this case is simply the friction on the skirts. The test at 100mm/s (but zero ambient water pressure) shows a larger capacity, and it is straightforward to show that this is controlled by cavitation beneath the foundation. This means that at elevated water pressures (as in the third test) the capacity rises approximately in step with to the ambient water pressure, as correspondingly larger pressure changes are required to cause cavitation. This problem is studied in more detail by Houlsby et al. (2005a).
It is important to note, however, that although ambient water pressure increases the ultimate capacity, it has negligible influence on the tensile load at which a flexible response begins to occur.
Comparison of cyclic loading tests at different scales and at different speeds shows that it is difficult to scale reliably the accumulated displacements, which reduce with larger tests and higher loading rates. However, when the scaling rules described earlier are applied, the shapes of individual hysteresis loops at different scales and at different rates become remarkably similar. Figure 33 shows a comparison, for instance, of loops at three different load amplitudes from four different tests. At each particular load amplitude the loops from the different tests are very similar.
The accumulation of displacement after very large numbers of cycles is difficult to predict, and so far few data are available. Rushton (2005) has carried out vertical loading tests to about 100000 cycles on a model caisson in sand, using a simple loading rig which employs a rotating mass and a series of pulleys to apply a cyclic load. A typical result is shown in Figure 34, on a caisson 200mm diameter and 100mm deep, with cycling between
N260210 ± . The caisson is therefore subjected (at the minimum vertical load) to a small tension, but less than the frictional capacity of the skirts. The dimensionless accumulated vertical displacement is seen in Figure 34 to increase approximately with the logarithm of the number of cycles of loading (after about 1000 cycles). Note that even in this case where there is a tensile loading in part of the cycle, the net movement is downwards. The displacement is of course very sensitive also to the amplitude of the cycling.
6.2 Clay: field and laboratory tests
Very few vertical loading tests relevant to the wind turbine problem have been completed on caissons in clay, although there have been a number of studies directed towards suction caissons used as tension
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0150 160 170 180 190 200 210
Vertical Displacement (mm)
Verti
cal S
tress
(kP
a)
5mm/s, 0kPa100mm/s, 0kPa100mm/s, 200kPa
Direction of movement
Figure 32: Tensile capacity of model caisson pulled at different rates and at different ambient pressures.
-1
0
1
2
3
4
5
0 0.01 0.02 0.03 0.04 0.05
[w/(2R)][pa/(2Rγ')]1/2
V/[ γ
'(2R
)3 ]
1.5m Field
0.15m Suction
0.2m Pushed
0.15m Pushed
Figure 33: Hysteresis loops from tests at different scales and rates.
anchors, e.g. El-Gharbawy (1998), Watson (1999), House (2002).
At Bothkennar tests were carried out in which inclined (but near vertical) loading was applied to a 1.5m diameter caisson (Houlsby et al., 2005b). Difficulties were encountered with the control of the loads using a hydraulic system, and the resulting load paths are therefore rather complex, leading to difficulties in interpretation. Further work on vertical loading in clay is required before definitive conclusions can be drawn, and in particular the issue of tensile loading in clay needs attention. Some preliminary results (Byrne and Cassidy, 2002), shown in Figure 35, show that the tensile response may be sensitive to prior compressive loading. Footings loaded in tension immediately after installation showed a stiff tensile response, whilst those loaded after first applying a compressive load to failure showed a more flexible tensile response.
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-40
-20
0
20
40
60
0 0.2 0.4 0.6 0.8 1
Normalised Displacement, (w + L)/D
Verti
cal S
tress
, V/A
(kPa
)
Test 1: Post Bearing Capacity
Test 2: Pre Bearing Capacity
Figure 35: Tension tests on caisson foundations in clay
7. NUMERICAL STUDIES
7.1. Finite element studies A number of analyses of suction caissons for offshore wind farms have been carried out as part of commercial investigations for possible projects. A
more detailed research project was carried out by Feld (2001).
Finite element analysis is particularly appropriate for establishing the effects of design parameters on the elastic behaviour of caissons, and has been used by Doherty et al. (2004a,b) to determine elastic stiffness coefficients for caisson design which take into account the flexibility of the caisson wall as well as coupling effects between horizontal and moment loading.
7.2 Plasticity models An important tool for the analysis of soil-structure interaction problems, particularly those involving dynamically sensitive structures are “force resultant” models. In these the behaviour of the foundation is represented purely through the force resultants acting upon it, and the resulting displacements (see Figure 4). Details of stresses and deformations within the soil are ignored. The models are usually framed within the context of work-hardening plasticity theory. Examples include models for foundations on clay (Martin and Houlsby, 2001) and on sand (Houlsby and Cassidy, 2002). Overviews of the development of these models are given by Houlsby (2003) and Cassidy et al. (2004)
These models have been further developed specifically for the offshore wind turbine application. The developments include: • Generalisation to full three-dimensional loading
conditions, • Inclusion of special features to represent the
caisson geometry, • Expression of the models within the “continuous
hyperplasticity” framework to allow realistic description of hysteretic response during cyclic loading.
A model with all these features is described by Lam and Houlsby (2005). The fitting of cyclic data to a continuous hyperplastic model is discussed by Byrne et al (2002a).
8. OTHER CONSIDERATIONS
We have concentrated here on the design of caisson foundations as far as capacity and stiffness are concerned for in-service conditions. However, there a number of other issues which need to be addressed in a caisson design, and we mention them here briefly.
8.1 Scour Scour is more important for caissons, since they are relatively shallow, than for piles. The size of caissons, and the fact that part of the caisson
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
1 10 100 1000 10000 100000
Number of Cycles
[w/(2
R)][
p a/(2
Rγ’)
]1/2
MinMax
Figure 34: Accumulated displacement during long term cyclic vertical loading on sand
inevitably protrudes above mudline level, creates rather aggressive conditions for scour. The fact that the caissons may be installed in mobile shallow-water environments means that proper consideration of this problem is essential, especially in sands.
If the scour depth can be determined with sufficient confidence (e.g. from comprehensive model testing) then it may be possible to permit the scour to occur, and simply allow for this in the design by ensuring that the caisson is deep enough.
It is more likely, however, that scour protection measures such as rock-dumping will need to be employed. Practical experience suggests that such protection must be placed very soon after caisson installation, as scour can occur very rapidly. In highly mobile environments, significant scour can, for instance, occur due to the currents in a single tide. Model testing indicates, however, that scour protection measures can be effective in preventing further erosion (R. Whitehouse: private communication). For in-service conditions regular monitoring for the possibility of scour would be prudent.
8.2 Liquefaction The transient pore pressures induced in the seabed can induce liquefaction, especially if the seabed is partially saturated due to the presence of gas (as can occur in shallow seabeds, largely due to decay of organic matter).
The problem is a complex one, but typically, at one stage in the wave cycle, the pore pressure in the seabed can become equal to the overburden stress, and the effective stress falls to zero. This problem is further complicated by the presence of a structure, which clearly modifies the pore pressure pattern that would occur in the far field. Although some progress has been made, the interactions are complex, and theoretical modelling of the problem is not straightforward.
8.3 Wave-induced forces A quite different problem from liquefaction is also related to the fact that the principal forces on the structure are wave induced. As a wave passes the column of the structure it exerts large horizontal forces (of the order of a few meganewtons for a large wave), which also cause overturning moments. However, at the same time the wave causes a transient pressure on the seabed, and on the lid of the caisson. Because the caissons are in shallow water these pressures are quite large. The pore water pressure within the caisson is unlikely to change as rapidly as the pressure on the lid, so there will be pressure differentials across the lid of the caisson
which result in net vertical forces, and overturning moments on the caisson.
The relative phase of the different sources of loading is important. As the crest of the wave just reaches the structure, the wave kinematics are such that the horizontal forces are likely to be largest. At this stage the pressure on the upwave side of the caisson is likely to be larger than on the downwave side. The net result is that the moment caused by the pressures on the caisson lid opposes that caused by the horizontal loading, so this effect is likely to be beneficial to the performance of the caisson. Little work has, however, yet been completed on the magnitudes of these effects. The problem is complicated by the fact that the kinematics of large (highly non-linear) shallow water waves is still a matter of research, as is their interaction with structures.
8. CONCLUSIONS
In this paper we have provided an overview of the extensive amount of work that has been carried out on the design of suction caisson foundations for offshore wind turbines. Further verification of the results presented here is still required, and in due course it is hoped that this will come from instrumented caisson foundations offshore. Our broad conclusions at present are: • Suction caissons could be used as foundations
for offshore wind turbines, either in monopod or tripod/tetrapod layout.
• The combination of low vertical load and high horizontal load and moment is a particular feature of the wind turbine problem.
• Stiffness and fatigue are as important for turbine design as ultimate capacity.
• Monopod foundation design is dominated by moment loading.
• Tripod/tetrapod foundation design is dominated by considerations of tensile loading.
• The moment-rotation response of caissons in sand has been extensively investigated by model tests and field trials, and modelled theoretically by finite element analyses and force resultant (yield surface) models.
• As amplitude of moment loading increases, stiffness reduces and hysteresis increases.
• Moment loading in clay has been less extensively investigated in the laboratory and field.
• Vertical loading in sand has been extensively investigated in the laboratory and field.
• The as the amplitude of vertical loading increases, stiffness reduces and hysteresis increases. Once tension is reached there is a sudden reduction of stiffness.
• Whilst high ultimate tensile capacities are possible (especially in deep water) this is at the expense of large movements.
• Application of scaling procedures for tests in both sand and clay allows model and field tests to be compared successfully as far as stiffness and the shapes of hysteresis loops is concerned.
• Cumulative displacements after very many cycles are harder to model.
• The design of caisson foundations also needs to take into consideration issues such as scour and liquefaction.
It is hoped that the conclusions above lead in due course to application of suction caissons as foundations for offshore wind turbines, thereby making an important renewable energy source more economically viable.
ACKNOWLEDGEMENTS
The work at Oxford University has been supported by the Department of Trade and Industry, the Engineering and Physical Sciences Research Council and a consortium of companies: SLP Engineering Ltd, Aerolaminates (now Vestas), Fugro Ltd, Garrad Hassan, GE Wind and Shell Renewables. An outline of the project is given by Byrne et al. (2002b). The work of Richard Kelly, Nguyen-Sy Lam and Felipe Villalobos on this project is gratefully acknowledged.
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13. Cassidy, M.J., Martin, C.M. and Houlsby, G.T. (2004) "Development and Application of Force Resultant Models Describing Jack-up Foundation Behaviour", Marine Structures, (special issue on Jack-up Platforms: Papers from 9th Int. Conf. on Jack-Up Platform Design, Construction and Operation, Sept. 23-24, 2003, City Univ., London), Vol. 17, No. 3-4, May-Aug., 165-193
14. Coldicott, L. (2005) “Suction installation of cellular skirted foundations”, 4th year project report, Dept. of Engineering Science, Oxford University
15. Doherty, J.P., Deeks, A.J. and Houlsby, G.T. (2004a) "Evaluation of Foundation Stiffness Using the Scaled Boundary Method", Proc. 6th World Congress on Computational Mechanics, Beijing, 5-10 Sept., in press
16. Doherty, J.P., Houlsby, G.T. and Deeks, A.J. (2004b) "Stiffness of Flexible Caisson Foundations Embedded in Non-Homogeneous Elastic Soil", Submitted to Proc. ASCE, Jour. Structural Engineering Division
17. El-Gharbawy, S.L. (1998) “The Pullout Capacity of Suction Caisson Foundations”, PhD Thesis, University of Texas at Austin
18. Feld T. (2001) “Suction Buckets, a New Innovative Foundation Concept, applied to offshore Wind Turbines” Ph.D. Thesis, Aalborg University Geotechnical Engineering Group, Feb..
19. Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999) "The Plastic Response of Circular Footings on Sand under General Planar Loading", Géotechnique, Vol. 49, No. 4, pp 453-470
20. Houlsby, G.T. (2003) "Modelling of Shallow Foundations for Offshore Structures", Proc. Int. Conf. on Foundations, Dundee, 2-5 Sept., Thomas Telford, pp 11-26
21. Houlsby, G.T. and Byrne, B.W. (2000) “Suction Caisson Foundations for Offshore Wind Turbines and Anemometer Masts”, Wind Engineering, Vol. 24, No. 4, pp 249-255
22. Houlsby, G.T. and Byrne, B.W. (2005a) “Design Procedures for Installation of Suction Caissons in Clay and Other Materials”, Proc. ICE, Geotechnical Eng., Vol. 158 No. GE2, pp 75-82
23. Houlsby, G.T. and Byrne, B.W. (2005b) “Design Procedures for Installation of Suction Caissons in Sand”, Proceedings ICE, Geotechnical Eng., in press
24. Houlsby, G.T. and Cassidy, M.J. (2002) "A Plasticity Model for the Behaviour of Footings on Sand under Combined Loading", Géotechnique, Vol. 52, No. 2, Mar., 117-129
25. Houlsby, G.T., Kelly, R.B. and Byrne, B.W. (2005a) "The Tensile Capacity of Suction Caissons in Sand under Rapid
Loading", Proc. Int. Symp. on Frontiers in Offshore Geotechnics, Perth, Australia, September, in press
26. Houlsby, G.T., Kelly, R.B., Huxtable, J. and Byrne, B.W. (2005b) “Field Trials of Suction Caissons in Clay for Offshore Wind Turbine Foundations”, Géotechnique, in press
27. Houlsby, G.T., Kelly, R.B., Huxtable, J. and Byrne, B.W. (2005c) “Field Trials of Suction Caissons in Sand for Offshore Wind Turbine Foundations”, submitted to Géotechnique
28. House, A. (2002) “Suction Caisson Foundations for Buoyant Offshore Facilities”, PhD Thesis, the University of Western Australia
29. Ibsen, L.B., Schakenda, B., Nielsen, S.A. (2003) “Development of bucket foundation for offshore wind turbines, a novel principle”. Proc. USA Wind 2003 Boston.
30. Ibsen, L.B. and Brincker, R. (2004) “Design of New Foundation for Offshore Wind Turbines”, Proceedings of The 22nd International Modal Analysis Conference (IMAC), Detroit, Michigan, 2004.
31. Ibsen L.B., Liingaard M. (2005) “Output-Only Modal Analysis Used on New Foundation Concept for Offshore Wind Turbine”, in preparation
32. Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M. (2003) "Pressure Chamber Testing of Model Caisson Foundations in Sand", Proc. Int. Conf. on Foundations, Dundee, 2-5 Sept., Thomas Telford, pp 421-431
33. Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M., 2004. Tensile loading of model caisson foundations for structures on sand, Proc. ISOPE, Toulon, Vol. 2, 638-641
34. Kelly, R.B., Houlsby, G.T. and Byrne, B.W. (2005a) "A Comparison of Field and Laboratory Tests of Caisson Foundations in Sand and Clay" submitted to Géotechnique
35. Kelly, R.B., Houlsby, G.T. and Byrne, B.W. (2005b) "Transient Vertical Loading of Model Suction Caissons in a Pressure Chamber", submitted to Géotechnique
36. Lam, N.-S. and Houlsby, G.T. (2005) "The Theoretical Modelling of a Suction Caisson Foundation using Hyperplasticity Theory", Proc. Int. Symp. on Frontiers in Offshore Geotechnics, Perth, Australia, Sept., in press
37. Martin, C.M. (1994) "Physical and Numerical Modelling of Offshore Foundations Under Combined Loads", D.Phil. Thesis, Oxford University
38. Martin, C.M. and Houlsby, G.T. (2000) "Combined Loading of Spudcan Foundations on Clay: Laboratory Tests", Géotechnique, Vol. 50, No. 4, pp 325-338
39. Martin, C.M. and Houlsby, G.T. (2001) “Combined Loading of Spudcan Foundations on Clay: Numerical Modelling”, Géotechnique, Vol. 51, No. 8, Oct., 687-700
40. Rushton, C. (2005) “Cyclic testing of model foundations for an offshore wind turbine”, 4th year project report, Dept. of Engineering Science, Oxford University
41. Sanham, S.C. (2003) “Investigations into the installation of suction assisted caisson foundations”, 4th year project report, Dept. of Engineering Science, Oxford University
42. Villalobos, F.A., Byrne, B.W. and Houlsby, G.T. (2005) "Moment loading of caissons installed in saturated sand", Proc. Int. Symp. on Frontiers in Offshore Geotechnics, Perth, Australia, Sept., in press
43. Villalobos, F., Houlsby, G.T. and Byrne, B.W. (2004) "Suction Caisson Foundations for Offshore Wind Turbines", Proc. 5th Chilean Conference of Geotechnics (Congreso Chileno de Geotecnia), Santiago, 24-26 November
44. Watson, P.G. (1999) “Performance of Skirted Foundations for Offshore Structures”, PhD Thesis, the University of Western Australia
45. Wolf, J.P. (1994) “Foundation Vibration Analysis Using Simple Physical Models”, Prentice Hall, New Jersey
1 INTRODUCTION
Shallow foundations are usually designed on the as-sumption that they act in isolation. When two foot-ings (or a group of footings) are closely spaced, however, there is a beneficial interaction that can be quantified in terms of the ‘efficiency’, i.e. the ratio of the overall (group) bearing capacity to the sum of the individual (isolated) bearing capacities. The lit-erature on this topic has been surveyed by Hazell (2004). For footings on sand, numerous theoretical and experimental studies have shown that the effect of interaction becomes highly significant for friction angles greater than about 30° and spacings less than about one footing width B. In contrast, the undrained bearing capacity of closely spaced footings on clay has received very little attention, perhaps because the early theoretical work of Mandel (1963) showed that the beneficial effect of interaction was insignificant, even for fully rough footings. This was confirmed experimentally by Hazell (2004), though only a few of his tests were conducted on clay.
When considering the relevance of these findings to the design of grillages or closely spaced footings on soft offshore soils, it is important to note that the theoretical studies by Mandel (1963) were confined to homogeneous soil, and although in the experi-ments of Hazell (2004) there was a marked increase of undrained strength with depth, the dimensionless ratio kB/su0 was no more than 0.2 for the small model footings tested (su0 = mudline strength inter-cept, k = rate of increase of strength with depth). In
water depths greater than a few hundred metres, the undrained strength at seabed level can be as low as 2 to 10 kPa, increasing with depth at 1 to 2 kPa/m (Randolph 2004). A typical offshore mudmat might be 5 m wide, and if supported by a single strip foot-ing (without perforations) this would imply typical values of the ratio kB/su0 in the range 0.5 to 5. When calculating the bearing capacity of an isolated foot-ing at the upper end of this range, the influence of non-homogeneity on the bearing capacity would cer-tainly be accounted for, either by adopting appropri-ate plasticity solutions (Davis & Booker 1973, Houlsby & Wroth 1983), or by selecting a represen-tative strength su > su0. It is therefore of interest to investigate the undrained bearing capacity of closely spaced footings for a similar range of kB/su0, and to assess the effect of using a mudmat with perforations in place of a continuous foundation (this is some-times done to save weight, and to make the structure easier to remove). For a small degree of perforation it might be envisaged that there will be arching over the gap(s), such that there is no loss of bearing ca-pacity, and even if some soil is squeezed through, there may still be a beneficial interaction effect. Here we investigate these issues using plasticity analyses.
2 BEARING CAPACITY ANALYSES
2.1 Isolated footings The bearing capacity of an isolated strip footing on non-homogeneous clay was first studied by Davis &
Bearing capacity of parallel strip footings on non-homogeneous clay
C.M. Martin & E.C.J. Hazell Department of Engineering Science, University of Oxford, UK
ABSTRACT: On soft seabed soils, subsea equipment installations are often supported by mudmat foundation systems that can be idealised as parallel strip footings, grillages, or annular (ring-shaped) footings. This paper presents some theoretical results for the bearing capacity of (a) two parallel strip footings, otherwise isolated; (b) a long series of parallel strip footings at equal spacings. The soil is idealised as an isotropic Tresca mate-rial possessing a linear increase of undrained strength with depth. The bearing capacity analyses are performed using the method of characteristics, and the trends of these (possibly exact) results are verified by a compan-ion series of upper bound calculations based on simple mechanisms. Parameters of interest are the footing spacing, the relative rate of increase of strength with depth, and the footing roughness. An application of the results to the design of perforated mudmats is discussed.
Booker (1973). They showed that for any value of kB/su0 (zero to infinity) the stress and velocity fields computed using the method of characteristics fur-nished lower and upper bounds that were coincident. Davis & Booker’s analyses have since been verified by several authors, and they can also be replicated using the free computer program ABC (Martin 2004). Figure 1a shows the variation of the bearing capacity factor Nc as a function of kB/su0. Note that Nc is defined with respect to the mudline strength, i.e. as Qu/Bsu0 where Qu is the ultimate bearing ca-pacity (per unit run). Figure 1b shows, for the same range of kB/su0, the extent of the zone of plastic de-formation adjacent to each side of the footing. This distance is important because it also corresponds to the critical spacing at which parallel strip footings first begin to interact and give an overall bearing ca-pacity that is greater than the sum of the individual capacities. As kB/su0 increases, the zone of plastic deformation becomes smaller, so the footings need to be closer for any interaction to occur.
Also shown in Figure 1 are the results of upper bound calculations performed using the generalised Hill- and Prandtl-type mechanisms of Kusakabe et al. (1986), which were originally developed for cir-cular footings on non-homogeneous clay. There is close agreement with the exact curves when kB/su0 is small, but this deteriorates somewhat with increasing non-homogeneity, especially for the rough footing.
2.2 Interacting footings – methodology In principle, bearing capacity analyses can be per-formed for an arbitrary number of parallel strip foot-ings at arbitrary spacings, but the calculations can become tedious, particularly when using the method of characteristics. Figure 2 shows the two problems considered here, both of which allow a favourable exploitation of symmetry: a pair of parallel strips, and an infinite number of parallel strips at equal spacings. Although impossible to realise in practice, the latter case is relevant to the interior members of a grillage containing many bearing elements. Note that in this paper S refers to the edge-to-edge spacing, not the centre-to-centre spacing as preferred by some au-thors. Note also that the footings are assumed to be rigidly connected, such that they move down to-gether without any horizontal displacement or rota-tion (for a pair of closely spaced footings there is a tendency for separation and tilting to occur).
The main bearing capacity analyses for interact-ing footings were performed using the method of characteristics. A modified version of the the Matlab program InterBC, developed by Hazell (2004) for in-teracting footings on a homogeneous c-φ-γ soil, was used. For a pair of footings, the program considers the right-hand footing and builds two meshes of characteristics, one commencing from the exterior soil surface, and one from the gap between the foot-
ing and the axis of symmetry (see Fig. 3). An itera-tive adjustment process is used to ensure that the two meshes are fully compatible at their common point C (same coordinates, same stresses). Having calculated the stress field, the program works back through the mesh and constructs the associated velocity field. These calculations are more complicated than those for an isolated footing, since the inward-flowing soil crosses a velocity discontinuity AD. Note that for non-homogeneous soil, the velocities outside AOD are not always parallel to the characteristics.
Two separate calculations of the bearing capacity are performed: one by integrating the stresses along ACB (deducting the self-weight of the ‘false head’ if applicable), and the other by equating the internal and external work rates of the collapse mechanism. In all of the analyses for this study it was found that, as the mesh of characteristics was refined, the two calculations of the bearing capacity converged to identical values. While this indicates that there are no regions of negative plastic work, it does not nec-essarily mean that the calculated bearing capacity is the exact collapse load, since it has not been demon-strated that the stress field can be extended through-out the soil. Construction of such an extension is straightforward for isolated footings (see e.g. Davis & Booker 1973, Martin 2005), but for interacting footings it would be necessary to incorporate a non-plastic zone to allow the major principal compres-sion to flip from horizontal to vertical at some point on the z axis. This is not easy, and suggests that the method of characteristics cannot readily be used to obtain strict lower bounds for interacting footings (finite element limit analysis could be used).
For a pair of smooth footings, the ‘squeezing’ type failure of Figure 3a is always critical (except on homogeneous soil, where there is no interaction ef-fect, and an infinite number of one- and two-sided mechanisms giving Nc = 2 + π can be devised). For a pair of rough footings that are very closely spaced, overall failure as a single footing of width 2B + S may be more critical than the squeezing failure of Figure 3b. When determining the variation of bear-ing capacity with spacing, it is always necessary to check this alternative mechanism, for which the bearing capacity can be determined using ABC.
Figure 4 shows some typical solutions for the case of infinitely many, equally spaced footings on non-homogeneous clay. Here the double symmetry means that it is only necessary to analyse half of one footing, so analysis using the method of characteris-tics is relatively straightforward. A single mesh is constructed, starting from the soil surface and ‘bouncing’ characteristics off the centerline of the gap as before. The mesh is adjusted until point C lies directly beneath the centre of the footing, with the major principal compression aligned vertically. Cal-culation of the associated velocity field – again in-corporating a discontinuity along AD – can then be
performed. As in the two-footing case, it was always found that the stress- and velocity-based calculations of the bearing capacity converged to the same value as the mesh was refined. This converged value represents a rigorous upper bound, but not a rigorous lower bound since the stress field is incomplete. If the number of footings is truly infinite then there is no alternative overall failure mechanism – squeezing failure is the only option, and the bearing capacity must approach infinity as the spacing tends to zero.
As well as the analyses performed using InterBC, some simple Hill- and Prandtl-type upper bound mechanisms were devised for the problems shown in Figure 2. These were based on the mechanisms of Kusakabe et al. (1986), but modified to allow a rigid wedge of soil to be extruded vertically between the interacting footings. In Figures 3 and 4 the outlines of the optimal upper bound mechanisms are super-imposed on the method of characteristics solutions, and there is a fairly close correspondence between the two. Note that for rough footings, the Prandtl-type mechanism shown in Figures 3b and 4b is only critical when kB/su0 is small; otherwise a Hill-type mechanism (similar to Figs 3a, 4a) governs.
2.3 Interacting footings – results Concentrating first on the pair of interacting foot-ings, Figure 5 shows the variation of efficiency (as defined at the start of the paper) with the normalised spacing S/B. For a spacing of zero, the two footings behave as a single footing of width 2B. Although this has no net effect when the soil is homogeneous, there is a beneficial interaction when the strength in-creases with depth since the influence of non-homogeneity is enhanced (2kB/su0 is greater than kB/su0, so the operative Nc is greater in Fig. 1a). For smooth footings the efficiency immediately begins to drop as soon as a gap is introduced, while for rough footings there is a brief increase in efficiency prior to the transition between overall and squeezing failure (see Section 2.2). In all cases there is a gradual de-cline towards unit efficiency as the critical spacing plotted in Figure 1b is approached.
The results for homogeneous soil (Fig. 5a) agree with those of Mandel (1963): there is no gain in effi-ciency for a pair of smooth footings, and a peak of just 1.07 (at S/B = 0.15) for a pair of rough footings. When the strength increases with depth, the potential gains in efficiency are rather greater, but the spacing needs to be small (< 0.1B), and the benefit from in-teraction is almost all attributable to the effective augmentation of kB/su0 rather than a genuine arching effect. In fact, the rough footing curves in Figure 5 clearly show that the influence of arching diminishes rapidly as non-homogeneity becomes more signifi-cant; when kB/su0 ≥ 2 there is an almost immediate transition from overall failure to squeezing failure as the spacing is increased from zero.
The predictions from the method of characteris-tics are consistent with those from the simple upper bound analyses, shown as dotted lines in Figure 5. The efficiency curve for a pair of rough footings on homogeneous clay (Fig. 5a) also agrees remarkably well with that obtained by Galloway (2004) using the finite element program ABAQUS. This suggests that the results obtained here from the method of characteristics may well be exact, though it is not immediately clear why the squeezing stress field should suddenly cease to become extensible at the same moment that overall failure becomes critical. It is noteworthy – and surely no coincidence – that Galloway’s analyses also predict a peak efficiency of 1.07 at a spacing of S/B = 0.15, coinciding with an abrupt transition from overall to squeezing failure.
Corresponding results for an infinite number of equally spaced footings are shown in Figure 6. The higher the value of kB/su0, the closer the footings need to be before there is any significant gain in effi-ciency (> say 10%). For a given spacing, the increase in efficiency is greatest for the homogeneous soil, and considerably higher (by a factor of up to 2) for rough footings than for smooth footings. The critical spacings at which the curves in Figure 6 reach unit efficiency are the same as those in Figure 5.
The dotted lines in Figure 6 show that the upper bound calculations give the same general trend, but they overpredict the efficiency factor quite seriously as the spacing becomes small. This is because the simple collapse mechanisms (consisting only of fan zones and rigid blocks) are unsuitable for modelling the increasingly complex velocity field as S/B → 0.
3 APPLICATION: PERFORATED MUDMATS
A common situation involving closely spaced foot-ings is the design of a mudmat with perforations. In practice such structures would usually have square or circular geometry, with the bearing elements tak-ing the form of a bidirectional grillage, or perhaps created by making a series of regularly spaced holes in an initially solid base. Although the actual failure mechanisms in these cases are complex (requiring 3D analysis), it is nevertheless instructive to perform some simplified calculations for plane strain condi-tions, to examine the general effect of introducing perforations. For brevity, only the two extreme cases shown in Figure 7 are considered. The notation adopted is the same as that used in the paper by White et al. elsewhere in these proceedings: W is the overall width of the mudmat, B* is the width of the individual bearing elements, and R is the perforation ratio (i.e. the fraction of W that has been removed). In Figure 7a, the ratios S/B* and kB*/su0 both change constantly as R is increased from zero (the former increases and the latter decreases). In Figure 7b, S/B* again increases as R is increased, but the ratio
kB*/su0 is effectively zero from the outset (since the number of perforations is very large, the width B* of each individual bearing element is very small). In both cases, as R is increased there eventually comes a point where there is no longer any interaction be-tween adjacent bearing elements of the mudmat.
Results for the single perforation scenario of Fig-ure 7a are shown in Figure 8. If the overall width W is taken as given, it is appropriate to characterise the non-homogeneity by kW/su0, and to define a ‘gross’ bearing capacity factor with respect to W, i.e. Nc = Qu/Wsu0. Using this convention, Nc for a smooth, singly-perforated mudmat on homogeneous soil decreases linearly from 2 + π to 0 as R increases from 0 to 1. The other curves in Figure 8a show that breaking W into two smaller widths causes most of the benefit derived from the increase of strength with depth to be lost quite quickly, followed by a more gradual decline once R passes about 0.1. The corre-sponding curves for rough-based mudmats with a single central perforation (Fig. 8b) have an initial plateau where overall failure is more critical than squeezing failure. However the beneficial effect of this arching across the perforation is only significant when kW/su0 is small.
Figure 9 presents results for the other scenario of a mudmat with numerous perforations (Fig. 7b). In this case both the smooth and rough curves exhibit plateaus corresponding to overall failure, but once squeezing failure becomes critical the bearing capac-ity starts to decline (somewhat more rapidly than in Fig. 8). Regardless of the value of kW/su0, the appro-priate squeezing curve is always that for homoge-nous soil, for the reason mentioned above: the ratio kB*/su0 is effectively zero because B* << W. (The curved envelopes in Figs 9a, b are simply alternative presentations of the data for kB/su0 = 0 in Figs 6a, b.) Perhaps the most interesting prediction in Figure 9 is that for a rough mudmat on homogeneous soil, a per-foration ratio in excess of 15% can be tolerated without any reduction in capacity. Unfortunately this is clearly not the case when kW/su0 > 0.
4 CONCLUSIONS
This paper has presented theoretical solutions for the vertical bearing capacity of rigidly connected, paral-lel strip footings on clay exhibiting a linear increase of undrained strength with depth. Both a pair of foot-ings and a large group of equally spaced footings have been considered. Results have been obtained using the method of characteristics, and confirmed by independent upper bound calculations based on simple mechanisms. The former results are believed to represent exact solutions, though they have only been established as upper bounds at this stage. A practical application to the design of perforated mudmats on soft offshore soil has been explored.
REFERENCES
Davis, E.H. & Booker, J.R. 1973. The effect of increasing strength with depth on the bearing capacity of clays. Géotechnique 23(4): 551-563.
Galloway, M. 2004. Interaction between adjacent footings in offshore foundation systems. Final year project report, De-partment of Engineering Science, University of Oxford.
Hazell, E.C.J. 2004. Interaction of closely spaced strip foot-ings. Final year project report, Department of Engineering Science, University of Oxford.
Houlsby, G.T. & Wroth, C.P. 1983. Calculation of stresses on shallow penetrometers and footings. OUEL Report No. 1503/83, Department of Engineering Science, University of Oxford.
Kusakabe, O., Suzuki, H. & Nakase, A. 1986. An upper-bound calculation on bearing capacity of a circular footing on a non-homogeneous clay. Soils and Found. 26(3): 143-148.
Mandel, J. 1963. Interférence plastique de fondations superfi-cielles. Proc. Int. Conf. on Soil Mech., Budapest: 267-280.
Martin, C.M. 2004. ABC – Analysis of Bearing Capacity. Software and documentation available for download from www-civil.eng.ox.ac.uk/people/cmm/software/abc.
Martin, C.M. 2005. Exact bearing capacity calculations using the method of characteristics. Issues lecture, Proc. 11th Int. Conf. of IACMAG, Turin, to appear.
Randolph, M.F. 2004. Characterisation of soft sediments for offshore applications. Keynote lecture, Proc. 2nd Int. Conf. on Site Investigation, Porto.
(a)
5
6
7
8
9
10
0 1 2 3 4 5
kB/su0
Nc =
Qu/B
s u0
SmoothRough
(b)
0
0.25
0.5
0.75
1
0 1 2 3 4 5
kB/su0
Cri
tical
S/B
SmoothRough
Figure 1. Isolated strip footing on non-homogeneous clay: (a) bearing capacity (b) critical edge-to-edge spacing for interac-tion between parallel footings. Results from simple UB calcula-tions (after Kusakabe et al. 1986) shown dotted.
(a)
su = su0 + kzφu = 0z
x
SB B
(b)
etc. etc.S S S SSB B BB
su = su0 + kzφu = 0
x
z Figure 2. Parallel strip footings on non-homogeneous clay: (a) pair of footings (b) many footings, equally spaced. (a) Smooth
(b) Rough
Figure 3. Pair of parallel strip footings on non-homogeneous clay (kB/su0 = 1, S/B = 0.15): characteristics and velocity vec-tors, with mechanism outlines from simple UB calculations. (a) Smooth
(b) Rough
Figure 4. Many parallel strip footings on non-homogeneous clay (kB/su0 = 1, S/B = 0.15): characteristics and velocity vec-tors, with mechanism outlines from simple UB calculations.
(a) kB/su0 = 0
1
1.1
1.2
1.3
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy
SmoothRough
Smooth effic. = 1 for all S/B
(b) kB/su0 = 1
1
1.1
1.2
1.3
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy
SmoothRough
(c) kB/su0 = 2
1
1.1
1.2
1.3
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy
SmoothRough
(d) kB/su0 = 5
1
1.1
1.2
1.3
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy
SmoothRough
Figure 5. Pair of parallel strip footings: variation of efficiency with edge-to-edge spacing. Efficiency = ratio of overall (group) capacity to sum of individual (isolated) capacities. Results from simple UB calculations shown dotted.
Note: half of one footing shown
Note: half of one footing shown
D
O A BC
C D
O A B
D
O A C
C D
O A
(a) Smooth
1
1.1
1.2
1.3
1.4
1.5
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy kB/su0 =0, 1, 2, 5
(b) Rough
1
1.1
1.2
1.3
1.4
1.5
0 0.1 0.2 0.3 0.4 0.5
S/B
Effi
cien
cy
kB/su0 =0, 1, 2, 5
Figure 6. Many parallel strip footings: variation of efficiency with edge-to-edge spacing. Efficiency = ratio of overall (group) capacity to sum of individual (isolated) capacities. Results from simple UB calculations shown dotted. (a)
W
B* S B*
Perforation ratio SRW
=
(b)
. . . . .
W
SB*
Perforation ratio *SR
B S=
+ Figure 7. Mudmat with (a) single central perforation (b) many identical perforations. Perforation ratio R = fraction of overall width W that has been removed.
(a) Smooth
0123456789
10
0 0.1 0.2 0.3 0.4 0.5
R = S/W
Nc =
Qu/W
s u0
kW/su0 =0, 1, 2, 5
(b) Rough
0123456789
10
0 0.1 0.2 0.3 0.4 0.5
R = S/W
Nc =
Qu/W
s u0
kW/su0 =0, 1, 2, 5
Figure 8. Mudmat with single central perforation: variation of bearing capacity with perforation ratio. Initial plateaus in (b) correspond to overall failure of mudmat. (a) Smooth
0123456789
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0 0.1 0.2 0.3 0.4 0.5
R = S/(B*+S)
Nc =
Qu/W
s u0
kB/su0 =0, 1, 2, 5
envelope continueslinearly to (1, 0)
kW/su0 =0, 1, 2, 5
(b) Rough
0123456789
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0 0.1 0.2 0.3 0.4 0.5
R = S/(B*+S)
Nc =
Qu/W
s u0
envelope continueslinearly to (1, 0)
kB/su0 =0, 1, 2, 5kB/su0 =0, 1, 2, 5kW/su0 =0, 1, 2, 5
Figure 9. Mudmat with many identical perforations: variation of bearing capacity with perforation ratio. Initial plateaus corre-spond to overall failure of mudmat.
1 INTRODUCTION 1.1 Motivation The response of shallow foundations subjected to general loading is an important area of civil engineering, particularly in the offshore industry, where foundations must be designed for loadings due to harsh environmental conditions. These conditions may lead to large vertical (V), horizontal (H) and moment (M) loads on the foundations. Whilst earlier studies considered overall stability, more recent studies have attempted to model the displacements, using model tests to calibrate work hardening plasticity theories (Houlsby et al., 1999; Martin and Houlsby, 2000, 2001; Byrne and Houlsby, 2001; Cassidy et al. 2002; Houlsby and Cassidy, 2002).
Recently, this work has focussed on suction caisson foundations (Byrne et al., 2002; Byrne and Houlsby, 2003). With geometry rather like an upturned bucket, the caisson is simply installed by sucking the water out, and thus forcing the skirts into the seabed. This type of foundation has potential applications in the developing offshore wind energy industry. In this application the loading consists of very high moment and horizontal loads, but low vertical loads. This is a very different pattern of loading from that experienced by heavier structures in the oil and gas sector. In addition, the wind and wave directions may not coincide, so the base shear and moment are not in the same direction. Considerable uncertainty surrounds how these
foundations may perform under these loading conditions (Byrne and Houlsby, 2003).
1.2 Background Theory Figure 1 shows a shallow foundation under three
degree-of-freedom loading as defined by Butterfield et al. (1997). This problem has received much attention over the past twenty years, and the load displacement behaviour of the foundation can be captured well by work-hardening plasticity theories (as shown by the papers cited above). A key component of the plasticity theories is the definition of a suitable yield surface. Figure 2 shows the shape of a yield surface that has been defined experimentally, for shallow foundations under three degree-of-freedom loading. This shape can be expressed mathematically as equation 1.
( ) 01
2
21 2212
22
=−−
−
+
=
βββ vv
mm
hha
mm
hhf
oooo (1)
where oV
Vv = , oRV
Mm2
= , oV
Hh = , ho is the
normalised horizontal load capacity, mo is the normalised moment capacity, a is the eccentricity of the ellipse in the h:m plane,
( )( ) 2
21
2112 21
21
+=+
ββ
ββ
βββββ and β1 and β2 are shaping
Investigating 6 degree-of-freedom loading on shallow foundations
B.W. Byrne & G.T. Houlsby Department of Engineering Science, University of Oxford, United Kingdom
ABSTRACT: Previous laboratory studies of the response of shallow foundations have only considered planar loading. This paper describes the development of a loading device capable of applying general loading on model shallow foundations. Loading involving all six degrees of freedom {vertical (V), horizontal (H2, H3), torsion (Q) and overturning moment (M2, M3)}, can be applied experimentally to the model foundations. Aspects of the design, including the loading rig configuration, development of a six degree-of-freedom load cell, numerical control algorithms and an accurate displacement measuring system are described. Finally results from initial experiments are presented that provide evidence for the generalisation of existing work-hardening plasticity models from planar loading to the general loading condition.
parameters for the section in the vertical load plane. Numerous studies have identified the parameter values for the yield surface for a variety of footing types and for different soils - for example see Houlsby et al. (1999) for shallow circular foundations on sand, or Martin and Houlsby (2000) for spudcans on clay. A natural extension of the these theories is to six degrees-of-freedom and Martin (1994) proposed an expression for this case:
( ) 012 21 2212
3223
222
23
23
22
=−β−
−−
+
+
+
+
=
ββ vvmh
mhmha
mm
mm
hh
hhf
oo
ooooo (2)
where oV
Hh 22 = ,
oVH
h 33 = ,
oRVMm
22
2 = ,
oRVM
m2
33 = and
oRVQq
2= . Figure 3 shows the
definitions of the loads from Butterfield et al. (1997). The displacements work-conjugate to the loads ( )3232 ,,,,, MMQHHV are ( )3232 ,,,,, θθωuuw . There has been no systematic study of footing response to full six degree-of-freedom loading to verify the extension of the planar loading theories to the general case. In the following the development of a loading device capable of applying the general loading is discussed, and some
initial experimental results are presented that can be used to verify equation 2.
2 DESIGN OF A 6 D-O-F LOADING RIG
2.1 The loading system Previous experimental work at Oxford has used a three degree-of-freedom (3dof) loading device designed by Martin (1994). This planar loading device achieves vertical and horizontal motion by using a system of sliding plates, and rotational movement by rotating the loading arm relative to these plates. All motions are independent of each other, and are each driven by a stepper motor – these features are useful for implementing load and displacement control systems. However, this type of system would become too cumbersome for six degree-of-freedom (6dof) motions, and so an alternative approach is required. Typically, in robotics applications, the Stewart Platform (Stewart, 1965) is considered to be the most elegant approach to achieving 6dof movement of a platform. There are numerous applications of this system in robotics, but the authors do not believe the system has been used for the testing of civil engineering structures, and in particular testing of foundations The arrangement described in this paper is a variant of the Stewart platform, and similar arrangements are used, for instance, in the automobile industry for dynamic testing of vehicles.
The system uses six actuators which, at one end, are connected to the loading platform, and at the other are connected to a stiff reaction frame. Provided that six properly arranged actuators are used, and are pinned at both ends, then it is possible to achieve 6dof motion of the loading platform by changing the lengths of the actuators in a co-ordinated fashion. By careful selection of the actuator geometry, it is possible to ensure that the control problem is well-conditioned, so that calculations proceed in a straightforward fashion.
The disadvantage with the Stewart Platform is that the simple motions are not linearly or independently related to the motion of any individual actuator, unlike the 3dof system of Martin (1994).
u
w
θ
Reference position
Current positionM
V
H
2R
Figure 1: Sign conventions for 3 degree-of-freedom loading(Butterfield et al., 1997).
V
H
M/2R
Yield surface Figure 2: Yield surface for shallow foundations.
V
2r
1
3
2
Q
M3
M2
H3
H2
Figure 3: 6dof loading on a circular foundation.
Therefore, quite complex control routines are required to ensure that all actuators move in concert to achieve the desired motion. Figure 4 shows the loading rig as constructed, showing three actuators approximately vertical and three actuators approximately horizontal. This arrangement ensures that the problem is well conditioned, as the main motions can be directly related to the motions of a sub-set of the actuators. For example, to achieve vertical movement the three vertical actuators must move the same distance, while only a slight adjustment of the horizontal actuators is required.
The actuators, supplied by Ultra Motion, are linear actuators each powered by an Animatics SmartMotor. This brushless DC servo-motor incorporates an integrated control system featuring a motion controller, encoder and amplifier. The actuators have a maximum extension of 200mm and can move at rates of up to 5mm/s. Commands to the actuators can specify relative motions, position, velocity or acceleration. The actuators are daisy-chained together and commands can be sent to individual actuators and then executed simultaneously with a global command. More importantly, a number of moves can be downloaded to on-board memory on the motors, and then executed according to a synchronised clock system common to all actuators. This makes it possible to execute complicated platform motions provided one can determine, in advance, a time history of the individual actuator motions required.
2.2 The control program A program has been written in Visual Basic to control the loading system. The program allows input of a sequence of moves in terms of the motions ( )3232 ,,,,, θθωuuw of the platform, known as the pose. These motions can be described in terms of a rotation and translation matrix (i.e. a transformation matrix). This matrix can be applied to the co-ordinates of the pinned connections of the actuators with the loading platform to produce a new set of co-ordinates for the platform in its new position. If the
co-ordinates of the other (fixed) ends of the actuators are known, then it is possible to determine the required lengths of each actuator. To move the platform to the new position simply requires extending/retracting each actuator to its required length. This calculation procedure is known as the inverse kinematics problem and is a simple analytical calculation.
The opposite calculation, called the forward kinematics problem, is not so straightforward, and requires a numerical solution. If the lengths of each actuator are known, then it is possible to calculate the new pose of the platform. Within the actuators are linear potentiometers that allow the user to determine the current length of the actuator, and therefore the pose of the platform. Both inverse and forward kinematics procedures are used within the software as shown in Figure 5.
A typical test proceeds by determining the initial platform pose using the forward procedure. The user then specifies a sequence of moves in terms of platform pose. These moves are broken into a series of small moves so that the non-linearity of motion of each actuator can be captured. The inverse procedure is used to calculate for each of the moves the required length of each actuator. A file of actuator lengths with time (position-time data) is recorded. The relevant data from this file are sent to each actuator, and each movement is executed simultaneously. An on-board buffering system allows moves to be downloaded to each actuator. The actuators themselves use sophisticated control processes to determine the velocity and accelerations required, so that the actuator reaches each position at the time required, thereby ensuring a smooth motion.
While the moves are being performed the control program logs the data. In particular the actuator lengths are recorded and the platform pose is calculated and displayed.
2.3 The load cell The load cell was constructed using a thin walled cylinder of radius r = 27.5mm, wall thickness t = 0.475mm and length 70mm. It was fabricated from Aluminium alloy with a Young’s Modulus of 72 GPa and a shear modulus of 27.1 GPa. The thin
Figure 4: Photos of the 6dof loading rig including a close-up of the small LVDT measurement system.
Actuator Lengths
A1 A2 A3 A4 A5 A6
Platform Pose
w u2 u3 ω
θ2
θ3
Forward Kinematics
Inverse Kinematics
Figure 5: Calculation procedures used in computer program.
walled section was machined from a larger block, leaving heavy end flanges. The transition from thin-walled section to flange was smoothed at an appropriate radius to minimise stress concentrations. A total of 32 strain gauges are fixed to the outer surface of the cylinder to measure the appropriate strains. Figure 6 shows the completed cell. The strain gauges were arranged in six Wheatstone bridge circuits, each corresponding to the measurement of a particular load component. Each circuit was fully compensated for temperature. Eight gauges were used for the vertical and torque circuits, and four gauges for the moment and horizontal load circuits. The cell was calibrated by applying known loads and measuring the output from all six circuits. By varying the loads one at a time, it is possible to determine components of the matrix X relating loads to voltages in the equation XFC = where C is the circuit output vector and F is the load vector. Figure 7 shows the results from the six circuits for changes in the vertical load. The slopes for these six curves represent the components of the part of the matrix relating to vertical load (i.e. the first column of the matrix X). Inverting X produces a six by six calibration matrix that can be incorporated into the control program, so that loads are calculated during the experiment. Note that the design of the circuits is such that the off-diagonal terms are small. This is indicated in Figure 7 where only one circuit is responsive to the change in applied load.
2.4 Small LVDT system One determination of the platform pose is by using the linear potentiometers within the actuators. This, however, provides only a coarse measurement of the platform pose. In particular there are issues of electrical noise and rig stiffness which have a significant impact on both the resolution and accuracy of this measurement. To achieve a more accurate determination of the foundation movement a system of small LVDTs (20mm range) are used. These are placed in a similar configuration to the actuators, but supported on a separate frame as shown in Figure 4. The program carries out the forward kinematics calculation to determine the pose of the platform, given the measured lengths of the LVDTs. This allows very fine resolution of the foundation movement to the order of a few microns (Williams, 2005).
3 EXPERIMENTAL RESULTS
Some preliminary experimental results on a 150mm diameter flat circular footing using only displacement control are presented here. At the time of writing load control routines were being developed and are anticipated to be implemented in the near future. The experiments were carried out on Leighton Buzzard 14/25 silica sand. This is a uniform sand with particle sizes ranging from 0.6mm to 1.18m. The maximum and minimum void ratios are 0.79 and 0.49 respectively. The sand was prepared in a loose state with a relative density estimated as 20%. Fuller details of the experimental work are reported by ap Gwilym (2004), Stiles (2004) and Williams (2005). The experiments were designed to determine the shape of the yield surface in the six dimensions. A number of ‘swipe tests’ were performed with various combinations of translations and rotations at a constant vertical displacement. The swipe test has been used extensively to determine the shape of yield surfaces, see Martin (1994), Gottardi et al. (1999), Martin and Houlsby (2000), Byrne and Houlsby (2001).
Figure 6: The 6 degree-of-freedom load cell.
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200 250 300 350 400 450 500
Vertical Load (N)
Circ
uit O
utpu
t (V)
C1C2C3C4C5C6
Figure 7: Calibration curves for the loadcell under verticalloading.
0
100
200
300
400
500
600
0 5 10 15 20 25 30
Vertical Displacement (mm)
Vert
ical
Loa
d (N
)
Figure 8: Typical vertical loading results.
3.1 Vertical loading Prior to carrying out the swipe tests it was necessary to perform vertical loading tests, as these give information for the hardening law. Five experiments are shown in Figure 8, showing good repeatability of the results. Note that the measurement of the displacement is coarse, as in these experiments the small LVDT system was not used.
3.2 Swipe tests A number of swipe tests were performed to investigate the suitability of equation 2. A typical experimental result for a swipe test is shown in Figure 9. In this test the footing was displaced vertically to a pre-specified distance at which point the vertical load reached approximately 530N. At this load the footing was translated horizontally. The figure shows that as the footing translates horizontally the relevant horizontal load traces a path around a yield surface. In this particular test the translation was u2 so the only horizontal load developed was H2. It is instructive to observe that the other load components are all relatively unaffected by the translation, as was expected. It is also possible to carry out tests involving translations u2, u3, -u2 and -u3. The results of these translations are shown in Figure 10 where the load paths for H2 and H3 are plotted. Note that each of the tests starts at a different vertical load. However, it is clear that the magnitudes and the shapes of the load paths are
similar for the different translations. This confirms the expectation that similar load paths will be traced out regardless of the translation direction. Similar experiments were carried out for rotations and twists with the same results (i.e. the results were independent of direction).
The data, such as shown in Figure 10, can be easily compared by normalising all the loads by Vo as suggested in equation 2. Results are plotted in Figure 11 for all possible pure horizontal, rotational and twisting swipes with the negative swipes reflected about the vertical load axis. It is clear that the results depend on the mode (i.e. translation/ twisting/rotation) of the swipe test but not on the direction. Equation 2 can be fitted to the above results to give the parameter values in Table 1, which are compared to data for footings on sand under planar loading.
Table 1: Parameter values for work-hardening model Parameter This
study Gottardi et al., 1999
Byrne and Houlsby, 2001
ho 0.122 0.122 0.154 mo 0.077 0.090 0.094 qo 0.033 N/A N/A β1 0.688 1.0 0.82 β2 0.709 1.0 0.82 a -0.212 -0.223 -0.25
In determining these parameters it was also
necessary to use results for combined swipes, that is swipes involving simultaneous rotation and translation and other combinations of movements. For instance Figure 12 shows the results from a test where a translation of u3 and rotation of -θ3 were applied simultaneously to the foundation. A number of these tests (twenty included in the above analysis) were performed as they are necessary in determining the fit, and in particular determining the parameter a which gives the rotation of the ellipse in the h:m plane. The test shown in Figure 12 could not have been performed using the previous 3dof loading rig as it involves non co-planar loads. Equally Figure 13 shows a test unique to the 6dof device in that during the swipe test the footing was first rotated by θ2 and
-80
-60
-40
-20
0
20
40
60
80
0 100 200 300 400 500 600
Vertical Load, V (N)
Hor
izon
tal L
oads
, H2,
H3 (
N)
Figure 10: Horizontal swipe results.
-20
-10
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600
Vertical Load, V (N)
Load
s, H
, M/2
R, Q
/2R
(N)
H2H3M2/2RM3/2RQ/2R
Figure 9: A horizontal swipe result.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
V /Vo
H/ V
o, M
/2R
V o, Q
/ Vo
Horizontal Swipes
Rotational Swipes
Twisting Swipes
Figure 11: Results normalised by Vo.
then rotated by θ3. (i.e. orthogonal and consecutive rotations). Initially under the rotation θ2 the load path for M2 tracks around a yield surface. When θ2 stops and θ3 starts the response for M2 drops off and the response for M3 picks up and eventually tracks around the same yield surface that M2 tracked.
4 CONCLUSIONS
In this paper the description of a unique loading device capable of applying six degree-of-freedom motion to a model foundation is presented. The resulting loads on the foundation are measured using a six degree-of-freedom load cell. A number of experiments, mainly displacement controlled swipe tests, are presented and interpreted to provide verification of the extension of a three degree-of-freedom plasticity model to six degrees-of-freedom. Further experimental work is required to verify the model fully.
5 ACKNOWLEDGEMENTS
The authors acknowledge the funding from the Lubbock Trustees (pilot project grant), the Royal Society (equipment grant), the Department of Engineering Science at Oxford University and EPSRC. We acknowledge the work of Clive Baker
in constructing the 6dof load cell and Chris Waddup who made the actuator frame and LVDT support frame. We also acknowledge the work carried out by final year undergraduate project students: Llywelyn ap Gwilym, Ed Stiles and Rachel Williams. The experimental work described here was conducted by these students under the direction of BWB.
6 REFERENCES
ap Gwilym, T.Ll. ab E. (2004). Control of a six degree of freedom loading rig. Fourth year project report, Department of Engineering Science, University of Oxford.
Butterfield, R., Houlsby, G.T. and Gottardi, G. (1997). Standardised sign conventions and notation for generally loaded foundations. Géotechnique 47, No 5, pp 1051-1054; corrigendum Géotechnique 48, No 1, p 157.
Byrne, B.W. and Houlsby, G.T. (2001). Observations of footing behaviour on loose carbonate sands. Géotechnique 51, No 5, pp 463-466.
Byrne, B.W. and Houlsby, G.T. (2003). Foundations for offshore wind turbines. Phil. Trans. Roy. Soc. A 361, Dec., pp 2909-2930.
Byrne, B.W., Houlsby, G.T., Martin, C.M. and Fish, P.M. (2002). Suction caisson foundations for offshore wind turbines. Wind Engineering 26, No 3.
Cassidy, M.J., Byrne, B.W. and Houlsby, G.T. (2002). Modelling the behaviour of a circular footing under combined loading on loose carbonate sand. Géotechnique 52, No 10, pp 705-712.
Gottardi, G., Houlsby, G.T. and Butterfield, R. (1999). The plastic response of circular footings on sand under general planar loading. Géotechnique 49, No 4, pp 453-470.
Houlsby, G.T. and Cassidy, M.J. (2002). A plasticity model for the behaviour of footings on sand under combined loading. Géotechnique 52, No 2, pp 117-129.
Martin, C.M. (1994). Physical and numerical modelling of offshore foundations under combined loads. DPhil Thesis, University of Oxford.
Martin, C.M. and Houlsby, G.T. (2000). Combined loading of spudcan foundations on clay: laboratory tests. Géotechnique 50, No 4, pp 325-338.
Martin, C.M. and Houlsby, G.T. (2001). Combined loading of spudcan foundations on clay: numerical modelling. Géotechnique 51, No 8, pp 687-700.
Stewart, D. (1965) A platform with six degrees of freedom. The Institution of Mechanical Engineers 180, No 15, pp371-384.
Stiles, E. (2004). Experiments using a six degree of freedom loading rig. Fourth year project report, Department of Engineering Science, University of Oxford.
Williams, R. (2005). Six degree of freedom loading tests on sand and clay. Fourth year project report, Department of Engineering Science, University of Oxford, in preparation.
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
V/Vo
Nor
mal
ised
Loa
ds, H
/Vo,
M/2
RVo
H3
M3/2R
Figure 12: Non co-planar loading applied to the foundation.
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1V /V o
M/2
RV o
M2/2RM3/2R
Figure 13: A swipe test where consecutive rotations are performed.
1. INTRODUCTION Suction caissons are an option for the foundations for offshore structures. Under large environmental loads the upwind foundations of a multiple-caisson foundation might be subjected to tensile loads. Recent research indicates that serviceability requirements will often dictate that, under working and frequently encountered storm loads, tensile loads on caissons should be avoided, as they are accompanied by large displacements. However, it may be appropriate to design structures so that under certain extreme conditions the caissons are allowed to undergo tension. It is therefore necessary to have a means of estimating the tensile capacity of a caisson foundation, whilst recognising that large displacements may be necessary to mobilise this capacity. The calculations are also relevant to the holding capacity of caisson anchors subjected to pure vertical load, and to calculation of forces necessary to extract a caisson rapidly (for whatever reason).
Under rapid tensile loading, a suction caisson in sand will exhibit a limiting load which will typically consist of a suction developed within the caisson, and friction on the outer wall. However, a number of different possible modes of failure exist. The purpose of this paper note is to set out simple calculations for capacities under various failure modes, and to compare these with experimental results.
2. TENSILE CAPACITY CALCULATIONS
2.1 Drained capacity If the tensile load is applied very slowly, then pore pressures will be small, and a fully drained calculation is applicable for calculating the capacity. For the purposes of calculation an idealised case of a foundation on a homogeneous deposit of sand is considered here.
The resistance on the caisson is calculated as the sum of friction on the outside and the inside of the skirt. The effective stresses on the annular rim are likely to be sufficiently small that they can be neglected, and it is assumed that the soil breaks contact with the lid of the caisson. The frictional terms are calculated in the same way as for the installation calculation (Houlsby and Byrne, 2005), by calculating the vertical effective stress adjacent to the caisson, then assuming that horizontal effective stress is a factor K times the vertical effective stress. Assuming that the mobilised angle of friction between the caisson wall and the soil is δ then we obtain the result that the shear stress acting on the caisson is δσ′ tanKv . Note that in the subsequent analysis the values of K and δ never appear separately, but only in the combination δtanK , so it is not possible to separate out the effects of these two variables. Allowance is made, however, for the possibility of different values of δtanK acting on the outside and inside of the caisson. A difference
The tensile capacity of suction caissons in sand under rapid loading
Guy T. Houlsby, Richard B. Kelly & Byron W. Byrne Department of Engineering Science, Oxford University
ABSTRACT: We develop here a simplified theory for predicting the capacity of a suction caisson in sand, when it is subjected to rapid tensile loading. The capacity is found to be determined principally by the rate of pullout (relative to the permeability of the sand), and by the ambient pore pressure (which determines whether or not the water cavitates beneath the caisson). The calculation procedure depends on first predicting the suction beneath the caisson lid, and then further calculating the tensile load. The method is based on similar principles to a previously published method for suction-assisted caisson installation (Houlsby and Byrne, 2005). In the analysis a number of different cases are identified, and successful comparisons with experimental data are achieved for cases in which the pore water either does or does not cavitate.
between this analysis and conventional pile design is that the contribution of friction in reducing the vertical stress further down the caisson is taken into account.
If, as a preliminary, no account is taken of the reduction of vertical stress close to the caisson due to the frictional forces further up the caisson, then the tensile vertical load on the caisson for penetration to depth h is given by:
( ) ( ) ( ) ( )iioo DKhDKhV πδγ′−πδγ′
−=′ tan2
tan2
22 (1)
friction on outside friction on inside
where the dimensions are as in Figure 1, and γ′ is the effective weight of the soil. V ′ is the buoyant weight of the caisson and structure.
A check should always be made that the friction calculated inside the caisson does not exceed the weight of the trapped soil plug 42
iDhπγ′ . Ignoring the reduction of the stress in this case
proves unconservative (i.e. it overestimates the force that can be developed), so we develop here a theory which takes this effect into account. Consider first the soil within the caisson. Assuming that the vertical effective stress is constant across the section of the caisson, the vertical equilibrium equation for a disc of soil within the caisson (Figure 2) leads to:
( ) ( ) ( )i
iv
i
iivvDK
D
DKdz
d δσ′−γ′=
π
πδσ′−γ′=
σ′ tan4
4
tan2
(2)
Writing ( )( ) iii ZKD =δtan4 , Eq. (2) becomes
γ′=σ′
+σ′
i
vvZdz
d, which has the solution
( )( )iiv ZzZ −−γ′=σ′ exp1 for 0=σ′v at 0=z . The total frictional terms depend on the integral of the vertical effective stress with depth, and we can also
obtain ( ) ( )( )iii
h
v ZhZhZdz +−−γ′=σ′∫ 1exp2
0. For
small iZh the integral simplifies to 22hγ′ as in Eq. (1). For brevity in the following we shall write the function ( ) ( )( )xxxy +−−= 1exp , so that in the
above ( )ii
h
v ZhyZdz 2
0γ′=σ′∫ .
A similar analysis follows for the stress on the outside of the caisson. We assume that (a) there is a zone between diameters oD and om mDD = in which the vertical stress is reduced through the action of the upward friction from the caisson, (b) within this zone the vertical stress does not vary with radial coordinate and (c) there is no shear stress on vertical planes at diameter mD . We then obtain the same results as for the inside of the caisson, but with
iZ replaced by ( ) ( )( )ooo KmDZ δ−= tan412 . Alternative assumptions could be made for the
variation of mD with depth, but at present there is little evidence to justify any more sophisticated approach. If mD is taken as a variable, then the differential equation for vertical stress will usually need to be integrated numerically.
Accounting for the effects of stress enhancement, Eq. (1) becomes modified to:
( ) ( )
( ) ( )ii
h
vi
oo
h
vo
DKdz
DKdzV
πδσ′
+πδσ′=′
∫
∫
tan
tan
0
0
(3)
In the special case where m is taken as a constant and uniform stress is assumed within the caisson this becomes:
( ) ( )
( ) ( )iii
i
ooo
o
DKZhyZ
DKZhyZV
πδ
γ′−
πδ
γ′−=′
tan
tan
2
2
(4)
σ'v(Ktanδ)iπDidz
σ'vπDi2/4
(σ'v + dσ'v)πDi2/4
γ'(πDi2/4)dz
Figure 2: Vertical equilibrium of a slice of soil within the caisson
z
Do
Di
h
V' Mudline
Figure 1: Caisson geometry
The calculation accounting for stress reduction obviates the need to check that the internal friction does not exceed the soil plug weight, as the capacity asymptotically approaches that value at large h.
2.2 Tensile capacity in the presence of suction If the caisson is extracted more rapidly, then transient excess pore pressures will occur, and the suction within the caisson will need to be taken into account. We return later to the calculation of the relationship between the rate of movement and the suction, but first address the calculation of load in terms of the suction. If the pressure in the caisson is s with respect to the ambient seabed water pressure, i.e. the absolute pressure in the caisson is
shp wwa −γ+ (where ap is atmospheric pressure,
wγ is the unit weight of water and wh the water depth), then we at first assume that the excess pore pressure at the tip of the caisson is as , i.e. the absolute pressure is ( ) ashhp wwa −+γ+ . There is therefore an average downward hydraulic gradient of
has wγ on the outside of the caisson and upward hydraulic gradient of ( ) hsa wγ−1 on the inside.
We assume that the distribution of pore pressure on the inside and outside of the caisson is linear with depth. A detailed flow net analysis shows that this approximation is reasonable. The solutions for the vertical stresses inside and outside the caisson are exactly as before, except that γ′ is replaced by
has+γ′ outside the caisson and by ( ) hsa−−γ′ 1 inside the caisson. The capacity, accounting for the pressure differential across the top of the caisson and pore pressure on the rim (only relevant for a thick caisson), is again calculated as the sum of the external and internal frictional terms:
( )
( ) ( ) ( ) ( )ii
h
vioo
h
vo
ioi
DKdzDKdz
DDas
DsV
πδσ′+πδσ′
=
−π+
π+′
∫∫ tantan
44
00
222
(5)
In the special case of m constant and a uniform stress assumed within the caisson, this gives:
( )
( ) ( )
( ) ( ) ( )iii
i
ooo
o
ioi
DKZhyZ
hsa
DKZhyZ
has
DDas
DsV
πδ
−−γ′−
πδ
+γ′−
=
−π+
π+′
tan1
tan
44
2
2
222
(6)
We can often make a further simplifying assumption, that the suction is sufficiently large that the soil within the caisson liquefies and therefore
( ) 01 =−−γ′h
sa . For a large suction this means that
1≈a and almost all of the suction appears at the caisson tip. The above rearranges to give
hs
has =+γ′ , and equation (6) can be simplified to:
( )
( ) ( )ooo
o
ioi
DKZhyZ
hs
DDas
DsV
πδ
−
=
−π+
π+′
tan
44
2
222
(7)
In the case either that the thickness of the caisson is small, or that 1≈a this simplifies to the following (writing the outer diameter as D, and the caisson area AD =π 42 ):
( )( )
( )
δ
+−=
−πδ
−=′
tan41
tan
2
2
KZhy
DhZsA
sADKZhyZ
hsV
(8)
where ( ) ( )δ−= tan412 KmDZ . Neglecting the effects of stress reduction would give:
( )
δ
+−=′ tan21 KDhsAV (9)
which means that the capacity is simply calculated by applying a linearly varying factor to the suction force beneath the lid.
2.3 Undrained failure A further condition should be considered: that of “undrained failure” of the sand. In any dilative sand, however, the pore pressures developed under undrained conditions are potentially so large that invariable (except in very deep water) the cavitation mechanism would intervene first. Since the undrained strength of sand is in any case very difficult to determine, we do not pursue this case here.
3. RELATIONSHIP BETWEEN SUCTION AND DISPLACEMENT RATE At low displacement rates, the rate of influx of water q to the caisson can be calculated by Darcy’s law, and equated to the rate of displacement times the
area of the caisson. Flow calculations were presented by Houlsby and Byrne (2005), and yield:
dtdhD
FDsk
q i
w
o4
2π−=
γ= (10)
where F is a dimensionless factor as determined by the procedures in Houlsby and Byrne (2005), which may be fitted approximately by the equation
( )DhF 516.3 += for 8.01.0 ≤≤ Dh . If the displacement rate is increased, the above
condition is interrupted by one of two conditions (a) the suction becomes large enough for liquefaction of the sand within the caisson to occur or (b) cavitation occurs within the caisson.
When liquefaction occurs, the permeability of the liquefied sand increases to a large value, with the result that the a factor in the calculation of the load changes (as noted above) to near unity. The displacement rate may still be estimated from a flow calculation, but the appropriate boundary condition now becomes one of the suction applied at the base rather than top of the caisson. Modified values of F (termed LF for this case) are given in Figure 3, and may be fitted approximately by the equation
( )DhF 5exp9.175.1 −+= for 0.11.0 ≤≤ Dh . When cavitation occurs, either before or after
liquefaction, the displacement rate becomes unlimited and (assuming that cavitation occurs at an absolute pressure afp where f is a constant), the suction will be constant and determined by
awwa fpshp =−γ+ , or ( ) wwa hfps γ+−= 1 . In practice it appears that the factor f is near zero.
4. SUMMARY OF ANALYSIS CASES The following summary presents equations for the above cases, for a thin-walled caisson. To simplify the equations we neglecting here the stress reduction effect, although this should be included in more
accurate calculations:
(a) Small dtdh−
dtdh
kD
Fs
o
wγπ−=4
(from Eq. (10)) and:
( ) ( ) ( )2
tan1tan2DhK
hsaK
has
sAV
ioπ
δ
−−γ′+δ
+γ′
−−=′
(for 0=− dtdh , 0=s and these reduce to the equations for the fully drained case).
(b) Liquefaction without cavitation
Onset of liquefaction occurs at ( )ahs
−γ′
=1
, after that
dtdh
kD
Fs
o
w
L
γπ−=4
and:
( )
δ
+−=′ oKDhsAV tan21
Note that this will imply a sudden jump in s and V ′ at the onset of liquefaction.
(c) Cavitation without liquefaction Onset of cavitation occurs at ( ) wwa hfps γ+−= 1 . After that dtdh− is unbounded, s is constant and:
( ) ( ) ( )2
tan1tan2DhK
hsaK
has
sAV
ioπ
δ
−−γ′+δ
+γ′
−−=′
as in case(a).
(d) Cavitation with liquefaction Since s is constant once cavitation occurs, this condition can only occur when liquefaction occurs before cavitation. Onset of cavitation is at
( ) wwa hfps γ+−= 1 , after which dtdh− is unbounded, s is constant and:
( )
δ
+−=′ oKDhsAV tan21 as in case (b).
Note that the above cases only occur in order (a), (b), (d) or (a), (c). When several possibilities exist for calculating load capacity it is often true that the correct case is simply found by calculating all cases and then taking the lowest value. Note in this analysis that this simple approach cannot be adopted as the onset of some states can preclude other cases occurring, and the calculated load is not necessarily the lowest of the cases.
5. COMPARISONS WITH DATA
We present here a number of pullout tests conducted two sands and at different pullout rates. The tests
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0 0.2 0.4 0.6 0.8 1.0Aspect ratio h/D
Dim
ensi
onle
ss fl
ow fa
ctor
FL
Figure 3: Dimensionless flow factor for liquefaction case
were conducted in a pressure chamber: some tests at an ambient (mudline) water pressure equal to atmospheric, and some at atmospheric plus 200kPa. The model caisson was 280mm diameter, 180mm skirt length. In the following the loads presented include the caisson weight.
The first test reported here (Test 9) was conducted on Redhill Sand, at a pullout rate of 100mm/s and atmospheric pressure. Figure 4 shows the record of suction developed beneath the lid of the caisson against time, and Figure 5 shows the corresponding vertical load. It can be seen that (with a minor initial fluctuation) the suction rapidly approaches 100kPa, at which stage cavitation occurs. At around 3044.5s there is a sudden loss of both suction and vertical load, but this is of little practical
interest since by then the displacements are enormous and about three-quarters of the caisson had been pulled out of the soil.
Figure 6 shows the ratio sAV / , showing that this ratio remains approximately constant during most of the pullout.
It can readily be shown that the suction in this case rapidly increased to sufficient value to cause liquefaction (which would occur at a suction of only about 3kPa), and that the relevant case for analysis here is case (d). The predicted values from the theory described above (including stress reduction) are also shown on each of Figures 4 to 6, and it is clear that the theory (whilst not capturing some of the detail at the beginning of the pullout) predicts the broad trends of the test correctly.
Figures 7 and 8 show corresponding results for Test 10 (at the same pullout rate) but at an ambient pressure of atmospheric plus 200kPa. The suctions developed at this rate of loading are insufficient to cause cavitation, which would occur at -300kPa relative to ambient. It can be seen that again the theory predicts the overall pattern of behaviour well. This time it is case (b) that applies. The fluctuations in predicted suction (and hence load) are due to minor variations in the calculated velocity of extraction.
Figures 9 and 10 show the results from Test 11, which is directly comparable to Test 9, but this time
-120.0
-100.0
-80.0
-60.0
-40.0
-20.0
0.0
20.0
3042.8 3043.3 3043.8 3044.3 3044.8 3045.3
t (s)
pres
sure
(kPa
)
ExperimentTheory
Figure 4: Pressure v. time for Test 9
-12.0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
3042.8 3043.3 3043.8 3044.3 3044.8 3045.3
t (s)
V (k
N)
ExperimentTheory
Figure 5: Vertical load v. time for Test 9
0.0
0.5
1.0
1.5
2.0
3042.8 3043.3 3043.8 3044.3 3044.8 3045.3t (s)
V / s
A
ExperimentTheory
Figure 6: V/sA v. time for Test 9
-300
-250
-200
-150
-100
-50
04257 4257.5 4258 4258.5 4259 4259.5
t (s)
pres
sure
(kPa
) ExperimentTheory
Figure 7: Pressure v. time for Test 10
-30.0
-25.0
-20.0
-15.0
-10.0
-5.0
0.0
5.0
4257 4257.5 4258 4258.5 4259 4259.5
t (s)
V (k
N)
ExperimentTheory
Figure 8: Vertical load v. time for Test 10
at a pullout rate of only 5mm/s. Although the suctions are sufficient to cause liquefaction, the pullout rate is such that the suction is sufficiently small so that cavitation does not occur, and the vertical loads are correspondingly lower too. The predicted suction and load are also shown on the Figures. The match to the data could be improved by adjusting the permeability, but the value used in the predictions were deliberately kept the same for all three tests discussed. The permeability value used was 3105.0 −×=k m/s, which is somewhat higher than estimated previously for this sand (Kelly et al. 2004). The other parameters used are 7.0tan =δK and 5.1=m .
Finally, Figures 11 and 12 present equivalent data for a test on HP5 sand, which is much finer that Redhill Sand, and has an estimated permeability of
4102.0 −×=k m/s. The extraction rate was 25mm/s, and in this case, although the extraction rate is lower, the pore pressures are sufficient to cause cavitation even with the ambient pressure of atmospheric plus 200kPa.
The predicted and measured values of maximum tensile load for the three tests on Redhill sand and one on HP5 are shown in Table 1. The order of magnitude of the tensile load is correctly predicted in all cases, even though the actual capacity of the caisson varies greatly in the different tests.
6. CONCLUSIONS
In this paper we develop a simplified theory for predicting the maximum tensile capacity of a caisson foundation in sand. The calculated capacity depends critically on the rate of pullout (in relation to the permeability) and the ambient water pressure (which determines whether cavitation occurs). The theory is used successfully to explain widely differing experimental results for caissons pulled out under different conditions.
REFERENCES Houlsby, G.T. and Byrne, B.W., 2005. Design procedures for
installation of suction caissons in sand, Proc. ICE, Geotechnical Engineering, in press.
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M., 2004. Tensile loading of model caisson foundations for structures on sand, Proc. ISOPE, Toulon, Vol. 2, 638-641
-20
-15
-10
-5
0
5
2980 2990 3000 3010 3020 3030
t (s)
pres
sure
(kPa
)
ExperimentTheory
Figure 9: Pressure v. time for Test 11
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
2980 2990 3000 3010 3020 3030
t (s)
V (k
N)
ExperimentTheory
Figure 10: Vertical load v. time for Test 11
-350
-300
-250
-200
-150
-100
-50
0
50
8890 8892 8894 8896 8898 8900 8902
t (s)
pres
sure
(kPa
)
ExperimentTheory
Figure 11: Pressure v. time for Test 23
-40
-30
-20
-10
0
10
8890 8892 8894 8896 8898 8900 8902
t (s)
V (k
N)
ExperimentTheory
Figure 12: Vertical load v. time for Test 23
Table 1: Predicted and measured tensile loadsMax. tensile load (kN)Test Predicted Measured
Test 11 (5mm/s, 0kPa) 1.1 2.4 Test 9 (100mm/s, 0kPa) 10.1 11.1 Test 10 (100mm/s, 200kPa) 25.6 24.2 HP5 sand: Test 23 (25mm/s, 200kPa) 30.6 33.2
1. INTRODUCTION In developments of offshore wind turbines, the foundations account for a significant fraction of the overall installed cost, approximately 15% to 40% of the total cost (Houlsby and Byrne, 2000). To satisfy the increasing need for renewable energy, there are a number of offshore wind farms to be constructed off the coast of the UK within the next few years. The possible use of caisson foundations for these turbines is therefore an important economical issue.
From previous research, there are elastic-plastic theoretical models available for analysis of shallow offshore foundations, such as “Model B” for a jack-up footing on clay (Martin, 1994) and “Model C” for footings on sand (Cassidy, 1999). These models are based on the idea of a “macro-element”, representing the foundation behaviour. The loading on the footing is represented by force resultants at a chosen reference point on the footing, and the movement by the corresponding displacements of this point. In this paper, a macro-element model for a caisson is presented in outline. The main goal of this work is to establish a theoretical framework to model correctly the cyclic behaviour of a caisson foundation, and this necessitates extension of previous modelling concepts to use of multiple yield surfaces.
2. CAISSON FOUNDATIONS A caisson foundation consists essentially of two parts: a circular top plate and a perimeter skirt, see Figure 1. The whole foundation is installed by the combination of gravity and suction within the caisson. In Figure 1, d is the distance between the
Load Reference Point (LRP) and an idealised soil surface position just as installation begins. The position of the LRP is arbitrary, but is conveniently taken at the joint between the caisson and the support structure. The conventions for forces are shown in Figure 2. The forces VR, H2R, H3R, QR, M2R, and M3R are applied at the LRP. In the analysis, however, it is often convenient to use the force system, ( )3232 , M, Q, M, HV, Hi =σ at the idealised soil surface level. The relationships between these two systems are: V = VR; H2 = H2R; H3 = H3R; Q = QR; M2 = M2R + dH3R; M3 = M3R – dH2R
The displacement vector at soil surface level is ( )3232 θθω=ε , , , , uw, ui . The corresponding
displacements at the LRP are given w = wR; u2 = u2R + dθ3R; u3 = u3R - dθ2R; ω = ωR; θ2 = θ2R; θ3 = θ3R.
3. RATE-INDEPENDENT SINGLE YIELD SURFACE HYPERPLASTICITY MODEL
Based on the hyperplasticity framework (Houlsby and Puzrin, 2000), a mechanical model can be derived from two scalar functions: the Gibbs free energy g, and either the dissipation function d or yield function y. The yield function is used here, since it can be identified directly from test results. The macro-element model is expressed in terms of the force vector iσ and displacement vector iε . It is also necessary to introduce the generalized force vector ( )3232 MMQHHVi , , , , , χχχχχχ=χ and finally the plastic displacement (or internal variable)
The theoretical modelling of a suction caisson foundation using hyperplasticity theory
Lam Nguyen-Sy & Guy T. Houlsby Department of Engineering Science, Oxford University
ABSTRACT: A theoretical model for the analysis of suction caison foundations, based on a thermodynamic framework (Houlsby and Puzrin, 2000) and the macro-element concept is presented. The elastic-plastic response is first described in terms of a single-yield-surface model, using a non-associated flow rule. To capture hysteresis phenomena, this model is then extended to a multiple yield surface model. The installation of the caisson using suction is also analysed as part of the theoretical model. Some preliminary numerical results are given as demonstrations of the capabilities of the model.
LRP
Soil surface level
d
mudline
2R
VR
HR MR
H
V
M
Figure 1. Geometry of caisson footing
vector ( )3232 MMQHHVi , , , , , αααααα=α . In general, for a model without elastic-plastic coupling, the free energy g and yield function y can be expressed as:
( ) ( )iiii ggg α+ασ−σ= 21 (1)
( ) 0,, =αχσ= iiiyy (2)
3.1 The Gibbs free energy function g In Eq. (1), g1 represents the elastic response of the foundation and is independent of plastic displacements. For linear elasticity it takes the following form:
DMHK
DMHK
DMK
DMK
KQ
DHK
DHK
KVg
324234233
223
5
2232
222
1
2
1
22
2222
−+−−
−−−−=
(3)
in which: K1 = 2GRk1; K2 = GR3k2 – 8GR2k4d + 2GRd2k3; K3 = 2GRk3; K4 = 4GR2k4 – 2GRk3d; K5 = 8GR3k5 and D = K2K3 – K4
2. G is the shear modulus of the soil, and the factors k1 … k5 are dimensionless stiffness coefficients as proposed by Doherty et al. (2004), who give elastic solutions for a caisson
foundation, in which the elastic stiffness of the caisson itself is taken into account.
The g2 term, which is the work of the plastic displacements, specifies the kinematic hardening of the model. A simple linear hardening relationship is achieved if g2 is a quadratic function of the plastic strains, and this form will later be used here as the basis for the multiple-surface model:
( )2332*4
23
*2
22
*2
2*52
3*3
22
*3
2*1
2
22
2222
HMHMMM
QHHV
HHH
HHHHg
αα−αα+α
+α
+
α+
α+
α+
α= (4)
where H1* … H5
* are hardening parameters which can conveniently be expressed in terms of the elastic stiffness factors K1 … K5. Details of these functions are discussed later. For the time being, however, we simply take 02 =g for the single surface model since the yield surface in this case does not undergo kinematic hardening.
3.2 The yield function y In the hyperplasticity framework (Houlsby and Puzrin, 2000), the yield function has been regconised as the singular Legendre transform of the dissipation function d in case of rate-independent materials. The yield surface is therefore expressed as a function of the generalised forces χi. Furthermore, the appearance of force components, (analogous to the true stresses σij in continuum plasticity models), in the form of the yield function leads automatically to non-associated flow rules (Collins and Houlsby, 1997), which are known as the appropriate to describe soil behaviour. Consequently, “association factors” which play an interpolation role between true and generalised forces are proposed in the yield function. Martin (1994) and Cassidy (1999) have establised the yield functions in elastic-plastic models (Model B and Model C for jack-up and circular footings of offshore structures on clay and sand). Based on these results, a yield function for a caisson footing is proposed with certain modifications to include aspects such as the non-associated flow rule:
01 2120112 =−+β−= ββ vtvSty (5)
in which:
( )( )[ ]201 1sgn vtvS −+= , 21
21
)()(
1
21
0
21
12 ββ
β+β
ββ
+β+β
=βt and
further definitions follow below:
V
VR
1
3
2
Q
M3
M2
2R
QR
M3R
M2RH3R
H2R
dH2
H3
Figure 2. Conventions for forces
( )233222
322
23
22 2 mhmhaqmmhht −+++++= (6)
( )( )0
111
1V
Vaav VVVV ρ−−+χ
= (7)
( ) ( )0
222
1V
Vaav VVVVV ρ−−+ρ+χ
= (8)
( ) ( )00
22222
1Vh
Haah HHHHH ρ−−+ρ+χ
= (9)
( ) ( )00
33333
1Vh
Haah HHHHH ρ−−+ρ+χ
= (10)
( ) ( )002
1
VRqQaa
q QQQQQ ρ−−+ρ+χ= (11)
( )( )( )( ) ( )
ρ+ρ−+−+ρ+χ+ρ+χ
=3232
3322
002 12
1
HMH
HHMMMddHMa
daVRm
m (12)
( )( )( )( ) ( )
ρ−ρ−−−+ρ+χ−ρ+χ
=2323
2233
003 12
1
HMH
HHMMMddHMa
daVRm
m (13)
It is convenient to note that the vertical load at which the maximum dimension of the yield surface is achieved
is 21
021
0 β+ββ−β
=α=t
VV , leading to α
αββ
−+
=10
2
1 t.
V0 is the vertical bearing capacity of the foundation (the intercept of the yield surface on the positive V-axis). Appropriate values of the parameters specifying the yield surface shape for a typical fully embedded caisson are β1 = β2 = 0.99; t0 = 0.1088. The parameters m0, h0 and q0 are factors which determine the sizes of the yield surface in the moment, horizontal and torsion directions: typical values are 0.15, 0.337 and 0.1 respectively.
The parameters aV1, aV2, aM, aH, aQ are the “association factors”; ρV, ρH2, ρH3, ρQ, ρM2 and ρM3 are the “back stresses” which are the difference between true force and generalised force, and are in turn expressed as functions of the internal variable
iα .
4. RATE-INDEPENDENT CONTINUOUS HYPERPLASTICITY MODEL
The main reason for the introduction of continuous hyperplasticity, which is in effect models an infinite number of yield surfaces, is to simulate a smooth transition between elastic and plastic behaviour, and capture with reasonable precision the hysteretic response of a foundation under cyclic loading. Such behaviour can not be described by a conventional single yield surface model.
4.1 The Gibbs free energy function g Starting from the form of Gibbs free energy function for a single yield surface model as in Eq. (1), further
developments for a continuous hyperplasticity model can be made. The Gibbs free energy function now becomes a functional as follows:
( )∫
∫∫∫
∫∫∫
∫
ηαα−αα+
ηα
+ηα
+ηα
+
α+η
α+η
α+
+ηασ−=
1
02332
*4
1
0
23
*2
1
0
22
*2
1
0
2*5
1
0
23
*3
1
0
22
*3
1
0
2*1
1
01
ˆˆˆˆˆ
2
ˆˆ
2
ˆˆ
2
ˆˆ
2
ˆˆ
2
ˆˆ
2ˆˆ
ˆˆ
dH
dH
dH
dH
Hd
Hd
H
dgg
HMHM
MMQ
HHV
ii
(14)
where η is a dimensionless parameter which varies from 0 to 1 and expresses the relative sizes of the yield surfaces. When η = 0, no plastic behaviour occurs. Once η = 1, fully plastic behaviour occurs. The hat notation is used to denote any function of η . The hardening parameters in Eq. (4) now become functions of η. These functions determine the shapes of the force-displacement curves. Hyperbolic curves may conveniently be used and for this case the hardening functions have the form:
( ) ( ) iniiii bKAH η−=η*ˆ (15)
where Ai, bi and ni are parameters defining the shape of the curves.
4.2 The yield function y For a certain value of η, the yield function can be expressed as follows:
0ˆ1ˆˆˆˆ 2120112 =−+−= βββη vtvSty (16)
where ( )2332
223
22
23
22 ˆˆˆˆ2ˆˆˆˆˆˆ mhmhaqmmhht −+++++= (17)
All the definitions of variables in Eqs. (16) and (17) are as for single-yield model, but these variables are determined for the yield surface corresponding to η.
5. MULTIPLE-YIELD SURFACE HYPERPLASTICITY MODEL
The concept of an infinite number of yield surfaces can model very well the response of a foundation under cyclic loading. However, to implement the model in a numerical analysis, it is necessary to discretise the continuous plasticity model to a multiple-yield-surface model. Firstly, the integrals in the Gibbs free energy become summations. Secondly, the continuously varying functions of η are replaced by discrete variables.
5.1 The Gibbs free energy function g The hat notation as in Eq. (14) is now abandoned to express the fact that the variables are no longer functions, but a series of discrete values. N is the number of yield surfaces chosen to simulate the continuous yield surface. We replace η by the factor j/N where j is the number of the yield surface which is being considered. In the summation, the increment dη in the integral becomes:
NNj
Njd 11 =−−=η (18)
The free energy function is therefore:
( )∑
∑∑∑
∑∑∑
∑
=
===
===
=
αα−αα+
α+
α+
α+
α+
α+
α+
+ασ
−=
N
jjHjMjHjM
j
N
j
jMjN
j
jMjN
j
Qjj
N
j
jHjN
j
jHjN
j
Vjj
N
j
iji
NH
NH
NH
NH
NH
NH
NH
Ngg
12332
*4
1
23
*2
1
22
*2
1
2*5
1
23
*3
1
22
*3
1
2*1
11
222
222 (19)
The above is appropriate provided that the N value chosen is large enough to result in a small dηi to achieve a reasonable approximation to the integral by use of a summation. Eq. (15) now becomes:
( ) ( ) ini
iii NjbKANjH −=
2* (20)
5.2 The yield function y Using the same style of yield function as in Eq. (16), the jth yield surface can be expressed as:
01 2120112 =−+β−=
ββjjjj vtvS
Njty (21)
Where equations exactly similar to (6) to (13) apply, but with each definition applying for the jth surface, thus Eq. (7) becomes for example:
( ) ( )( ) 0
111
1VNj
Vaav VjVVjVjV
jρ−−+ρ+χ
= (22)
The factors, S, β12, β1, β2 and t0, have the same values as in the single yield surface model. The definitions of the generalised forces can be expressed as follows:
Vj
n
VV
VjVj
V
Njb
KAVgN α
−−=α∂∂−=χ
21 (23)
jM
n
MM
jH
n
HH
jHjH
M
H
NjbKA
NjbKAHgN
3343
2232
22
2
3
2
2
2
α
−+
α
−−=α∂∂−=χ (24)
jM
n
MM
jH
n
HH
jHjH
M
H
NjbKA
NjbKAHgN
2242
3333
33
3
2
3
2
2
α
−−
α
−−=α∂∂−=χ (25)
Qj
n
QjQj
Q
Njb
KAQgN α
−−=α∂∂−=χ
25 (26)
jH
n
HH
jM
n
MM
jMjM
H
M
NjbKA
NjbKAMgN
3343
2222
22
2
3
2
2
2
α
−−
α
−−=α∂∂−=χ (27)
jH
n
HH
jM
n
MM
jMjM
H
M
NjbKA
NjbKAMgN
2242
3323
33
3
2
3
2
2
α
−+
α
−−=α∂∂−=χ (28)
The coordinates of center of jth yield surface in stress space can be defined as:
VjVj V χ−=ρ (29)
and likewise for the other variables. Figure 4 shows the form of yield surfaces after a
purely vertical loading. The size of the smallest yield surface in the vertical load direction is set as a certain fraction of the size of the outer yield surface. Between the inner and outer surfaces a uniform distribution of sizes of yield surfaces is used. The purpose of using a non-zero size of the first yield surface on the V-axis is to control the development of vertical plastic displacement on vertical unloading.
5.3 Incremental response In the multiple-yield-surface model, using rate-independent behaviour, the loading point must always be within or on each yield surface. This condition requires that the y-values for all active yield surfaces must be identically zero. The imposition of these “consistency conditions” is not straightforward in numerical analyses. The use of rate-dependent behaviour has been proposed as a means to simplify the numerical difficulties by Houlsby and Puzrin (2002) and Puzrin and Houlsby (2003). The dissipation function d is in this case separated into two functions; force potential function z and flow potential w. Houlsby and Puzrin (2002) note that w can take alternative forms, depending on
v = V/V0
t
t0
Initial fraction = 0.8
1.0
Figure 4. Multiple yield surfaces
the form of the viscosity assumed. In this paper, linear viscosity is used and the flow potential functions can be defined as:
( ) ( )µ
χασ=χασ
2
,,,,
2ijijij
ijijijy
w (30)
Where µ is the viscosity; jy is the jth yield function which no longer needs to be identically zero. Note that jw is only zero when the rates of change of plastic displacements are all zero. The incremental changes of plastic displacements caused by the jth yield surface can be defined as:
dtyy
dtw
dij
jj
ij
jij χ∂
∂µ
=χ∂
∂=α (31)
The total displacement increments are now calculated as:
∑= χ∂
∂µα∂σ∂
∂−σσ∂σ∂
∂−=εN
j ij
jj
ijik
kii dt
Njyyggd
1
22 (32)
6. NUMERICAL ILLUSTRATIONS Firstly, a result modelling the suction assisted penetration process using the concepts of Houlsby and Byrne (2005) is shown in Figure 5.
Secondly, a numerical example is given to illustrate test results which are obtained from laboratory testing of model caissons. In this example, AV = 1.0; AH2 = AH3 = AQ = AM2 = AM3 = 0.5; bV = bH2 = bH3 = bQ = bM2 = bM3 = 1.0; nV = nH2 = nH3 = nQ = nM2 = nM3 = 3.0. Twenty yield surfaces are used. The values of yield function parameters are: am = ah = 0.7; aV1 = 0.297; aV2 = 1.0; t0 = 0.1088; m0 = 0.15; h0 = 0.337; the shear modulus of the soil is G = 0.7MPa, self-weight γ = 15.74kN/m2, Poisson ratio ν = 0.2; initial fraction for the first yield function = 0.8. The radius of caisson R =
146.5mm; the length of the perimeter skirt H = 146.5mm. The caisson is installed to the full penetration position and then the horizontal and moment loads are applied. The vertical load V increases to the value of V = 945N during the penetration process and decreases to the value of V = 50N before the lateral loads are applied. During the
Figure 6. Rotation under cyclic loading
-20-15-10-50
5101520
-0.004 -0.002 0 0.002 0.004
theta3 (rad)
M3
(Nm
)
test results
theoretical results
Figure 7. Horizontal displacement under cyclic loading
-60
-40
-20
0
20
40
60
-0.0004 -0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003
u2 (m)
H2
(N)
test results
Theoretical results
Figure 5. Installation processes with and without suction
0
50
100
150
200
250
300
0 0.5 1 1.5 2
Vertical penetrations (m)
Verti
cal lo
ads
(kN
)
Vertical penetration w ithoutsuction
Vertical penetration w ithsuction assistance
application of the cyclic loads, the vertical load is kept constant at 50N. The resulting moment-rotation behaviour is shown in Figure 6, and horizontal load-displacement in Figure 7. Finally the vertical movements during the cycling are presented in Figure 8. In each case the analyses are compared to test results, and it can be seen that a satisfactory agreement is achieved.
7. DISCUSSION
There are four main points that must be addressed in this model: the choice of the hardening functions, the values of the association factors, the effects of suction pressures and the use of the rate dependent solution.
Firstly, the hardening functions, Hi*, determine
the distributions of plastic displacements which are caused by each yield surface. Therefore, the solutions can become stiffer or softer by increasing or decreasing the factors Ai, bi or ni. It is very important to determine the appropriate value of the shear modulus G. Since the hardening functions depend on the elastic stiffness factors, which include the shear modulus, the value of G strongly affects the solutions.
Secondly, the association factors play the role of determining the direction of the flow vectors of the plastic strains. To choose suitable values for these factors, it is necessary to consider some special aspects of the yield yield functions, such as the positions of the “parallel points” where the vertical plastic displacement incerments are zero. The directions of the flow vectors in the (V, M) plane, (M, H) plane or (V, H) plane can be obtained from tests. Furthermore, during the application of lateral loads, the upward or downward movements of the footing are also depend on the position of the parallel point and the value of the vertical load. However, the details of these expressions are beyond the scope of this paper.
Thirdly, as shown in Figure 5, by using suction, the vertical load which must be applied for
installation is rather small compared with that for installation using purely vertical load. This feature is very useful because it is impossible to apply a large value of vertical load to install the caisson in practice. Consequently, by using suction assisted penetration, this obstacle can be overcome.
Lastly, in order to avoid numerical difficulties, the rate-dependent solution has been proposed. The most important aspect of using the rate-dependent solution is the relationship among the viscosity µ, the time step dt and the load step. Suitable values must be chosen to maintain accuracy and stability for the numerical solution. There are as yet no precise procedures for selecting for these
parameters. However, by using some trials, one can determine suitable values for µ, dt and load step.
8. CONCLUSIONS
This paper presents a multiple yield surface hyperplasticity model for caisson foundations. Preliminary choices for the parameters are made. The model captures reasonably well the behaviour of a caisson foundation under cyclic loading, and could be incorporated in numerical analyses of caisson/structure systems.
REFERENCES Cassidy, M.J., 1999. Non-linear analysis of Jack-Up structures
subjected to random waves. DPhil thesis, University of Oxford.
Collins, I.F. and Houlsby, G.T., 1997. Application of Thermomechanical Principles to the Modelling of Geotechnical Materials. Proc. Royal Society of London, Series A, Vol. 453, pp 1975-2001
Doherty, J.P., Deeks, A.J. and Houlsby, G.T., 2004. Evaluation of Foundation Stiffness Using the Scaled Boundary Method. Proc. 6th World Cong. on Comp. Mech., Beijing, 5-10 Sept.
Houlsby, G.T. and Byrne, B.W., 2000. Suction caisson foundations for offshore wind turbines and anemometer masts. Wind Engineering, Vol. 24, No. 4, pp. 249-255.
Houlsby, G.T. and Byrne, B.W., 2005. Design Procedures for Installation of Suction Caissons in Sand, Proc. ICE, Geotechnical Engineering, in press.
Houlsby, G.T and Cassidy, M.J., 2002. A plasticity model for the behaviour of footings on sand under combined loading. Géotechnique, 52, No 2, pp. 117-129
Houlsby, G.T. and Puzrin, A.M., 2000. A Thermomechanical Framework for Constitutive Models for Rate-Independent Dissipative Materials. Int. J. of Plasticity, Vol. 16, No. 9, pp 1017-1047.
Houlsby, G.T. and Puzrin, A.M., 2002. Rate-Dependent Plasticity Models Derived from Potential Functions. J. of Rheology, Vol. 46, No. 1, Jan./Feb., pp 113-126.
Martin, C.M., 1994. Physical and numerical modelling of offshore foundations under combined loads. DPhil thesis, University of Oxford.
Puzrin, A.M. and Houlsby, G.T., 2003. Rate Dependent Hyperplasticity with Internal Functions, Proc. ASCE, J. Eng. Mech. Div., Vol. 129, No. 3, March, pp 252-263
Figure 8. Vertical movements under cyclic loading
0.14370.14380.14390.144
0.14410.14420.14430.14440.1445
-0.004 -0.002 0 0.002 0.004
theta3 (rad)
w (m
)
test results
theoretical
Figure 1: Dimensions and magnitude of loads for a 3.5MW turbine structure founded on a monopod suction (adapted from Byrne and Houlsby, 2003)
1. INTRODUCTION Suction caisson foundations are increasingly being used in offshore applications. They have been used for fixed structure applications, as described by Bye et al. (1995), and also for floating facilities (House, 2002). More recently they are being considered as foundations for offshore wind turbines (Byrne and Houlsby, 2003). The wind turbine structures may be founded on single or multiple caissons. The multiple caisson problem is addressed by Kelly et al. (2004), so in this paper we concentrate on the single caisson problem. Typical dimensions and loads for this problem are shown in Figure 1. Byrne and Houlsby (2003) describe this problem in detail, but the main differences in loads on the foundations for offshore wind turbines as compared to typical oil and gas structures are that: (a) the vertical load is much smaller, (b) the horizontal and moment loads are proportionately larger. New design methods must be developed to allow safe designs to be engineered for this regime of loading. As a result Byrne et al. (2002) describe a research project aimed at developing such design guidelines. This paper outlines the results from a part of that project.
Initial studies of the moment capacity of caisson foundations in the laboratory were carried out in drained sand. Preliminary results from these experiments are described by Byrne et al. (2003). As the sand used during the tests was dry, the caissons were installed into the prepared sand bed by applying a vertical load. The advantage of using dry sand is that the test bed can be prepared quickly, and a large number of tests can be carried out at specified densities. To mitigate the effects of scale,
the tests beds were chosen to be relatively loose. Clearly using installation by applying vertical loads is different from the procedure that has to be used in the field i.e. the suction installation process. The different installation techniques may impose different stress paths on elements of soil around the caisson, which in turn may affect the response of the caisson to the applied loads. Therefore it is necessary to carry out experiments similar to those
Moment loading of caissons installed in saturated sand
Felipe A. Villalobos, Byron W. Byrne & Guy T. Houlsby Department of Engineering Science, Oxford University
ABSTRACT: A series of moment capacity tests have been carried out at model scale, to investigate the effects of different installation procedures on the response of suction caisson foundations in sand. Two caissons of different diameters and wall thicknesses, but similar skirt length to diameter ratio, have been tested in water-saturated dense sand. The caissons were installed either by pushing or by using suction. It was found that the moment resistance depends on the method of installation.
in the dry sand, but on caissons installed by suction, to observe if there are any fundamental differences in behaviour.
Combined vertical, moment and horizontal loading tests have been conducted on caissons installed by suction and by vertical load in a water-saturated, dense sand. Load-displacement data are presented and interpreted for installation and for moment loading tests.
2. EQUIPMENT AND MATERIALS 2.1 Sand samples The sand used during the experiments was a commercially produced sand called Redhill 110. The properties of this sand are given in Table 1.
Table 1: Redhill 110 properties (Kelly et al., 2004)
D10, D30, D50, D60 (mm) 0.08, 0.10, 0.12, 0.13
Coefficients of uniformity, Cu and curvature Cc
1.63, 0.96
Specific gravity, Gs 2.65 Minimum dry density, γmin (kN/m3) 12.76 Maximum dry density, γmax (kN/m3) 16.80 Critical state friction angle, φcs 36º
The sand samples were saturated with water inside a tank of diameter 1100mm and depth 400mm. Preparation of the test bed involved an initial phase of fluidisation by an upward hydraulic gradient induced in the sand bed. The sample was then densified by vibration under a small confining stress. The density was determined by measuring the weight and the volume of the sample. The preparation process was halted once a target density was reached. The peak triaxial angle of friction was estimated as 44.1o to 45.2o from the correlation of Bolton (1986), for the range of relative densities tested (see Table 3).
2.2 Testing procedure Tests were performed using a three degree-of-freedom loading rig (3DOF) designed by Martin (1994). This rig, shown in Figure 2, can apply any combination of vertical, rotational and horizontal displacement (w, 2Rθ, u) to a footing by means of computer-controlled stepper motors (R is the radius of the footing). Byrne (2000) has installed a software control program, so that any combination of vertical, moment or horizontal load (V, M/2R, H) can also be applied to the footing. All displacements and loads are monitored and recorded using appropriate data-acquisition routines as well as being used within feedback control routines. It is possible to apply
loads and displacements to the footing which represent the offshore environment loads of gravity, wind, waves and currents. The geometry of the model suction caissons used in the experiments is given in Table 2. The model caissons were fabricated from aluminium alloy, with a relatively smooth (but not polished) surface.
Table 2: Geometry of the model caissons tested
Diameter, 2R (mm) 293 200 Length of skirt, L (mm) 146.5 100 Thickness of the skirt wall, t (mm) 3.4 1.0 Aspect ratio, L/2R 0.5 0.5 Thickness ratio, 2R/t 86 200
The loading apparatus was modified to allow the footings to be suction installed. Previous experiments had only used caissons forced into the ground by vertical load. To enable the suction installation phase to be carried out, the equipment was modified as shown in Figure 3. The suction caisson, attached to the 3DOF loading rig, was pushed into the ground about 30mm with the air valve open. This allowed the pressure inside the
Figure 2: 3DOF-loading rig
Figure 3: Suction device
caisson to equilibrate to the outside pressure. On reaching a penetration of 30mm the air valve was closed, and the fluid valve opened. The fluid from inside of the caisson was connected to a reservoir, which was slowly lowered to increase the head difference, hf, between the inside and outside of the caisson. The head difference was increased to a maximum of 300mm (3kPa), whilst the vertical load applied to the footing was kept constant using feedback control. The reservoir was connected to the suction caisson by a pipe of 6mm internal diameter, chosen to allow sufficient water flow with minimal head loss.
This procedure allowed the caisson to be installed by suction whilst connected to the loading rig. Once the suction phase was complete, it was possible to carry out experiments similar to those carried out on the dry sand as described by Byrne et al. (2003).
2.3 Comments on Installation Methods The two different installation methods have been described by Houlsby and Byrne (2005). Installation by vertical load involves pushing the caisson into the ground. The resistance to penetration is given by the friction on the inside and outside of the wall, and the bearing resistance on the skirt tip. Due to ‘silo effects’ the stresses around the skirt, and at the tip, are enhanced, leading to larger resistances than may be given by a more conventional pile calculation. Houlsby and Byrne (2005) developed expressions for predicting the resistance to penetration for caissons, taking account of these ‘silo’ effects. The equations they developed are used below to provide a comparison with the experimental results. In these calculations it will be assumed that (Ktanδ)i,o takes a value of 0.9, and the stress enhancement factors m and n are taken as 1.
Installation by suction requires an initial penetration to create a seal at the skirt tip. Typically 10% to 20% of the caisson skirt penetrates into the ground under its own weight. The seal allows the suction process to begin and should prevent the occurrence of an unconfined flow failure (i.e. a piping failure). Once sealed, the caisson will penetrate into the ground under the application of suction. Typically a pump will remove fluid from inside the caisson, creating a pressure differential on the caisson lid, as well as inducing hydraulic gradients in the soil. The hydraulic gradients lead to changes in the effective stresses around the caisson skirt that are beneficial to installation. Houlsby and Byrne (2005) have developed expressions to calculate the required suction for installation of
caissons. This expression is used below for comparison with the experimental data.
4. RESULTS OF THE INSTALLATION TESTS
Figure 4 shows the load-displacement results for caissons pushed into the ground at a constant rate. The results are shown as vertical load V against vertical penetration h. The maximum values of V obtained during these tests were approximately 400N for the footing of diameter 200 mm and 1700N for the 293 mm diameter footing. These maximum values of V (denoted by Vo in Table 3) represent “preconsolidation” vertical loads which might be used for interpreting the results within the context of a yield surface model (Gottardi et al., 1999; Houlsby and Cassidy, 2002). dR in Table 3 is the Relative Density. Also shown on Figure 4 is a theoretical prediction calculated using the methods of Houlsby and Byrne (2005).
Table 3: Installation tests (Suction and Pushing)
Test 2R mm
V N
Vo N
Rd %
h& mm/s
FV6_5_1S FV7_5_1P
200 200
5 -
398 425
75 74
0.04 0.50
FV6_2_1S FV6_3_1P
200 200
40 -
410 428
75 75
0.04 0.50
FV6_8_1S FV7_1_1P
200 200
60 -
400 469
75 74
0.04 0.50
FV8_1_1S FV8_2_1P
293 293
10 -
1700 1772
81 81
0.04 0.20
FV7_3_1S FV7_4_1P
293 293
60 -
1700 1740
74 74
0.06 0.20
FV7_1_3S FV7_2_1P
293 293
120 -
1500 1741
74 74
0.07 0.40
0200400600800
100012001400160018002000
0 20 40 60 80 100 120 140Vertical Displacement (skirt penetration), h (mm)
Verti
cal L
oad,
V (N
)
Theory (Houlsby and Byrne, 2005)Experiments, D = 293mmExperiments, D = 200mm
Figure 4: Pushing installation tests and theoretical calculations for both caissons
Figure 5 shows the load penetration curves for a pushed test and a suction-installed test. In the latter the vertical load was kept constant at 60 N during the suction phase. For this test the curve labelled V+S shows the net vertical load due to applied load plus the pressure differential on the caisson lid. It is clear that there is a significant difference between this net vertical load and the vertical load for the caisson installed by pushing. The difference between these curves represents the beneficial effects of the hydraulic gradients set up within the soil due to the suction.
Figure 6 shows one of these tests compared to the theoretical predictions of Houlsby and Byrne (2005). In all of the experimental tests the suction was applied after approximately 30mm of penetration. The suction force shown in Figure 6 is slightly underestimated by the calculations.
5. MOMENT CAPACITY
Once the caissons were installed, moment capacity tests were carried out. These tests are similar to
those reported by Byrne et al. (2003), and consist of rotation and translation of the footing at a specified ratio of M/2RH under a constant vertical load. The tests were carried out slowly, so that the conditions were fully drained. They are thus relevant to only one of a series of possible conditions in the field, where, depending on caisson size, sand type and loading rate, conditions may vary from fully drained to almost undrained. Summary data for the moment tests are presented in Table 4, and further data about initial conditions can be found in Table 3 (installation method, Vo, initial Rd, etc.)
Figures 7 and 8 compare the moment and horizontal load capacities for different installation methods for the 200mm diameter caisson. It is clear from these figures that the installation method has a strong effect on the load-displacement behaviour. The load-displacement curves have been interpreted using the method described by Byrne et al. (2003), with fitting linear expressions to the elastic and plastic components of the curve. The intersection of the lines represents a yield point. These points are shown on the figure and given in Table 4 for all the tests.
0
10
20
30
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Rotational Displacement, 2R( theta) (mm)
Mom
ent L
oad,
M/2
R (N
)
Pushed installation
Suction installation
Figure 7: Moment capacity tests, load-rotation response showing yield points (M/2R)y
-400
-300
-200
-100
0
100
200
300
0 20 40 60 80 100 120 140
Vertical Displacement (skirt penetration), h (mm)
Ver
tical
Loa
d, V
and
S (N
)
Test FV7_3_1: VCalculated VTest FV7_3_1: SCalculated S
Figure 6: V and S comparison between experimental result and calculation for a suction installed test of the 293mm diameter caisson
0
10
20
30
0 0.25 0.5 0.75 1
Horizontal Displacement, u (mm)
Hor
izon
tal L
oad,
H (N
)
Pushed installation
Suction installation
Figure 8: Two moment capacity tests, load-displacement response showing yield points Hy
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 20 40 60 80 100 120 140
Vertical Displacement (skirt penetration), h (mm)
Ver
tical
Loa
d, V
and
V +
S (N
)
by pushing FV7_4_1
by suction FV7_3_1 V = 60N
V
V
V +S
Figure 5: Comparison between pushed installation and suction installation for 293mm diameter caisson
The displacements paths from the tests are shown in Figures 9 and 10. In general the rotational displacement causes an initial elastic response that gradually changes to an almost perfectly plastic response, which can be fitted with a straight line.
The values of the slopes of these plastic displacement increments are presented in Table 4 as a ratio between horizontal and rotational displacement increments θ&& Ru 2/ and between vertical and rotational displacement increments
θ&& Rw 2/ . Figure 10 illustrates the change of vertical
displacement during the rotation of the caisson. The suction installed caisson experiences a lower magnitude of uplift compared with the caisson installed by pushing.
5.1 Yield Surface and velocity vectors Using the yield points ((M/2R)y, V) in Table 4, a plot of the yield surface for low vertical loads is illustrated in Figure 11. It is possible to fit through these data points a surface such as expressed by the formula:
( )( ) 011
22
2
21
21
22212
22
=⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
β
−⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
ββ
β+β oo
oo
oooooooo
VVt
VV
t
VRmM
VhHa
VRmM
VhHy
(1)
in which 1,,,, βatmh ooo and 2β define the shape of
the surface an ( )21
21
)()( 21
2112 ββ
β+β
ββ
β+β=β .
Table 4: Moment capacity tests
Test RHM
2V N yR
M⎟⎠⎞
⎜⎝⎛
2
N
Hy N θ&
&
Ru
2
θ&&
Rw
2
FV6_5_2SFV7_5_2P
1.03 1.02
5.56
6.7 14.3
4.8 12.8
0.3910.490
-0.397 -0.445
FV6_2_2SFV6_3_2P
1.06 1.05
40 40
11.8 24.1
10.4 21.7
0.4630.465
-0.122 -0.284
FV6_8_2SFV7_1_2P
1.05 1.03
60 60
18.3 29.2
16.4 26.9
0.5010.505
-0.051 -0.253
FV8_1_2SFV8_2_2P
1.04 1.03
10 10
14.8 33.4
15.1 32.9
0.3100.569
-0.409 -0.551
FV7_3_2SFV7_4_2P
1.04 1.03
60 60
30.9 42.0
27.7 40.4
0.4040.446
-0.299 -0.483
FV7_1_4SFV7_2_2P
1.04 1.03
120120
39.7 56.3
40.4 53.3
0.3620.477
-0.125 -0.289
The surface is fitted through the data points using parameter values given in Table 5. These values were found for a series of rotational tests performed with the same 293mm diameter caisson in dry sand. Also shown on Figure 11 are the directions of the displacement increment vectors.
The data for both footing diameters can be presented on the same figure by normalising with
0
10
20
30
40
50
60
-40 -30 -20 -10 0 10 20 30 40 50 60Vertical Load, V (N) - dw
Mom
ent L
oad,
M/2
R (N
), 2R
d(th
eta)
suction installedpushing installed
Figure 11: Pushing and suction installation calculations for a 200mm diameter caisson and V = 5N, 40N and 60N
0
0.25
0.5
0.75
1
1.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Rotational Displacement, 2R( theta) (mm)
Hor
izon
tal D
ispl
acem
ent, u
(mm
)by suction FV6_2_2 :V = 40N; D = 200mmby pushing FV6_3_2: V = 40N; D = 200mm
Figure 9: Plastic displacements increments during the tests: horizontal displacement with respect to rotational displacement
-1
-0.75
-0.5
-0.25
0
0.250 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Rotational Displacement, 2R( theta) (mm)
Verti
cal D
ispl
acem
ent, w
(m
m) by suction FV6_2_2 :V = 40N; D = 200mm
by pushing FV6_3_2: V = 40N; D = 200mm
Figure 10: Plastic displacements increments during the tests: vertical displacement with respect to rotational displacement
respect to Vo, the maximum applied vertical load. These results are shown on Figure 12. Equation (1) has been included in this plot with a value of to = 0.064 for the smaller footing and 0.040 for the larger footing. It is necessary to use different values of to in this plot because the tensile capacity scales with 2RL2 whilst the Vo value scales principally with
RtL2 . Since the two footings have the same RL 2/ value but different Rt 2/ values their tensile capacities differ on the normalised plot. However, the normalisation by Vo merges the two curves shown in Figure 11 for any one caisson, thus suction or pushing installation has only a minor effect on the normalised curve.
In more detail, however, the yield surfaces presented in Figure 12 serve as lower bounds for the moment capacity in the case of a caisson installed by pushing. On the other hand, it represents an upper bound for a suction installed caisson. The differences are thought to be due to disturbance in the installation process due to suction.
The incremental plastic displacement vectors were also compared. The vectors for suction installation tests have a smaller component in the w-direction compared with the vectors for pushed installation tests (see last column in Table 4 for values). Therefore, there was less uplift during the rotation of a caisson when a suction installation procedure was used.
Table 5: Yield surface parameters for L/2R = 0.5
Eccentricity of yield surface, a -0.75 Horizontal dimension of yield surface, ho 0.337Moment dimension of yield surface, mo 0.122Curvature factor at low V, β1 0.99 Curvature factor at high V, β2 0.99
6. CONCLUSIONS
A series of experiments comparing the moment response for suction installed caissons and those installed by pushing have been carried out. The main results are: (a) The use of suction beneficially reduces the
resistance to penetration of the caisson. (b) The moment resistance of a suction caisson
depends on the method of installation. (c) The ratio of horizontal and rotational plastic
displacement increments, pp Ru θ&& 2/ , was independent of the installation method.
(d) Under rotations more vertical uplift was observed for the pushed installed caisson than the suction installed caisson although this was also dependent on the applied vertical load.
(e) The yield surface (equation (1)) was applied successfully to two different size suction caissons after normalisation by Vo, but requires differing values of to, to account for different ratios of 2R/t.
REFERENCES Bolton, M.D. (1986) The strength and dilancy of sands.
Géotechnique, Vol 36, No. 1, pp65-78 Bye, A., Erbrich, C. T., Rognlien, B., and Tjelta, T. I. 1995.
Geotechnical design of bucket foundations. Offshore Technology Conference, Houston, paper 7793
Byrne, B.W. 2000. Investigations of suction caissons in dense sand, DPhil thesis, University of Oxford
Byrne, B.W. and Houlsby, G.T. 2003. Foundation for offshore wind turbines, Phil. Trans. of the Royal Society of London, Series A 361, 2909-2300
Byrne, B.W., Houlsby, G.T., Martin, C.M. and Fish, P.M. 2002. Suction caisson foundations for offshore wind turbines. Wind Engineering, Vol. 26, No 3.
Byrne, B.W., Villalobos, F.A., Houlsby, G.T. and Martin, C.M. 2003. Laboratory Testing of Shallow Skirted Foundations in Sand, Proc. Int. Conf. on Foundations, Dundee, 2-5 September, Thomas Telford, 161-173.
Gottardi, G., Houlsby, G.T. and Butterfield, R. 1999. The Plastic Response of Circular Footings on Sand under General Planar Loading, Géotechnique, 49, 4, pp 453-470.
Houlsby, G.T. and Cassidy, M.J. 2002. A plasticity model for the behaviour of footings on sand under combined loading, Geotéchnique, Vol. 52, No. 2, pp 117-129
Houlsby, G.T. and Byrne, B.W. 2005. Calculation procedures for installation of suction caissons in sand. Proc ICE, Geotechnical Engineering, in press
House, A.R. 2002. Suction Caisson Foundations for Bouyant Offshore Facilities, PhD thesis, the University of Western Australia
Kelly, R.B., Byrne, B.W., Houlsby, G.T. and Martin, C.M. 2004. Tensile loading of Model Caisson Foundations for Structures on Sand, Proc. ISOPE. Conf., Toulon
Martin, C.M. 1994. Physical and Numerical Modelling of Offshore Foundations under Combined Loads, DPhil thesis, University of Oxford
0
0.05
0.1
0.15
-0.1 -0.05 0 0.05 0.1 0.15Vertical Load, V/Vo , dw
Mom
ent L
oad,
(M/2
RV
o),
2Rd(
thet
a)suction installed ; D = 293 mmpushing installed ; D = 293 mmsuction installed ; D = 200 mmpushing installed ; D = 200 mm
Figure 12: Summary of experimental yield points (normalized by Vo) and incremental plastic displacement vectors
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